{"id":241,"date":"2025-04-09T17:33:25","date_gmt":"2025-04-09T17:33:25","guid":{"rendered":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/6-4-graphs-of-logarithmic-functions-college-algebra-2e-openstax\/"},"modified":"2025-08-29T17:28:06","modified_gmt":"2025-08-29T17:28:06","slug":"6-4-graphs-of-logarithmic-functions","status":"publish","type":"chapter","link":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/6-4-graphs-of-logarithmic-functions\/","title":{"raw":"6.4 Graphs of Logarithmic Functions","rendered":"6.4 Graphs of Logarithmic Functions"},"content":{"raw":"<div id=\"main-content\" class=\"MainContent__ContentStyles-sc-6yy1if-0 NnXKu\" tabindex=\"-1\" data-dynamic-style=\"true\">\r\n<div id=\"page_44418435-ed46-454a-aba4-cd57f5266654\" class=\"chapter-content-module\" data-type=\"page\" data-book-content=\"true\">\r\n<div class=\"textbox textbox--learning-objectives\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Learning Objectives<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nIn this section, you will:\r\n<ul>\r\n \t<li>Identify the domain of a logarithmic function.<\/li>\r\n \t<li>Graph logarithmic functions.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<p id=\"fs-id1165135194555\">In Graphs of Exponential Functions, we saw how creating a graphical representation of an exponential model gives us another layer of insight for predicting future events. How do logarithmic graphs give us insight into situations? Because every logarithmic function is the inverse function of an exponential function, we can think of every output on a logarithmic graph as the input for the corresponding inverse exponential equation. In other words, logarithms give the <em data-effect=\"italics\">cause<\/em> for an <em data-effect=\"italics\">effect<\/em>.<\/p>\r\n<p id=\"fs-id1165137603580\">To illustrate, suppose we invest [latex] \\$2500 [\/latex] in an account that offers an annual interest rate of [latex] 5\\% [\/latex] compounded continuously. We already know that the balance in our account for any year t can be found with the equation [latex] A=2500e^{-.05t}. [\/latex]<\/p>\r\n<p id=\"fs-id1165137668181\">But what if we wanted to know the year for any balance? We would need to create a corresponding new function by interchanging the input and the output; thus we would need to create a logarithmic model for this situation. By graphing the model, we can see the output (year) for any input (account balance). For instance, what if we wanted to know how many years it would take for our initial investment to double? Figure 1 shows this point on the logarithmic graph.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_959\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-959\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-1-300x159.webp\" alt=\"\" width=\"300\" height=\"159\" \/> Figure 1[\/caption]\r\n<p id=\"fs-id1165135161452\">In this section we will discuss the values for which a logarithmic function is defined, and then turn our attention to graphing the family of logarithmic functions.<\/p>\r\n\r\n<section id=\"fs-id1165137923503\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Finding the Domain of a Logarithmic Function<\/h2>\r\n<p id=\"fs-id1165137748716\">Before working with graphs, we will take a look at the domain (the set of input values) for which the logarithmic function is defined.<\/p>\r\n<p id=\"fs-id1165137758495\">Recall that the exponential function is defined as [latex] y=b^x [\/latex] for any real number <em>x<\/em> and constant [latex] b&gt; 0, b\\not=1, [\/latex] where<\/p>\r\n\r\n<ul id=\"fs-id1165137736024\">\r\n \t<li>The domain of <em>y<\/em> is [latex] (-\\infty, \\infty). [\/latex]<\/li>\r\n \t<li>The range of <em>y<\/em> is [latex] (0, \\infty). [\/latex]<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1165135641666\">In the last section we learned that the logarithmic function [latex] y=\\log_b(x) [\/latex] is the inverse of the exponential function [latex] y=b^x. [\/latex] So, as inverse functions:<\/p>\r\n\r\n<ul id=\"fs-id1165137656096\">\r\n \t<li>The domain of [latex] y=\\log_b(x) [\/latex] is the range of [latex] y=b^x: (0, \\infty). [\/latex]<\/li>\r\n \t<li>The range of [latex] y=\\log_b(x) [\/latex] is the domain of [latex] y=b_x: (-\\infty, \\infty). [\/latex]<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1165135245571\">Transformations of the parent function [latex] y=\\log_b(x) [\/latex] behave similarly to those of other functions. Just as with other parent functions, we can apply the four types of transformations\u2014shifts, stretches, compressions, and reflections.<\/p>\r\n<p id=\"fs-id1165137653624\">In Graphs of Exponential Functions we saw that certain transformations can change the <em data-effect=\"italics\">range<\/em> of [latex] y=b^x. [\/latex] Similarly, applying transformations to the parent function [latex] y=\\log_b(x) [\/latex] can change the <em data-effect=\"italics\">domain<\/em>. When finding the domain of a logarithmic function, therefore, it is important to remember that the domain consists <em data-effect=\"italics\">only of positive real numbers<\/em>. That is, the argument of the logarithmic function must be greater than zero.<\/p>\r\n<p id=\"fs-id1165137851584\">For example, consider [latex] f(x)=\\log_4(2x-3). [\/latex] This function is defined for any values of <em>x<\/em> such that the argument, in this case [latex] 2x-3, [\/latex] is greater than zero. To find the domain, we set up an inequality and solve for <em>x<\/em>.<\/p>\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{lll} 2x-3 &amp;&gt;&amp; 0 &amp;&amp; \\text{Show the argument greater than zero.} \\\\ 2x &amp;&gt;&amp; 3 &amp;&amp; \\text{add 3.} \\\\ x &amp;&gt;&amp; 1.5 &amp;&amp; \\text{Divide by 2.} \\end{array} [\/latex]<\/p>\r\n<p id=\"fs-id1165137645047\">In interval notation, the domain of [latex] f(x)\\log_4(2x-3) [\/latex] is [latex] (1.5, \\infty). [\/latex]<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Given a logarithmic function, identify the domain.<\/strong>\r\n<ol>\r\n \t<li>Set up an inequality showing the argument greater than zero.<\/li>\r\n \t<li>Solve for <em>x<\/em>.<\/li>\r\n \t<li>Write the domain in interval notation.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 1: Identifying the Domain of a Logarithmic Shift<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWhat is the domain of [latex] f(x)=\\log_2(x+3)? [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>The logarithmic function is defined only when the input is positive, so this function is defined when [latex] x+3&gt;0. [\/latex] Solving this inequality,\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{lll} x+3 &amp;&gt;&amp; 0 &amp;&amp; \\text{The input must be positive.} \\\\ x &amp;&gt;&amp; -3 &amp;&amp; \\text{Subtract 3.} \\end{array} [\/latex]<\/p>\r\nThe domain of [latex] f(x)=\\log_2(x+3) [\/latex] is [latex] (-3, \\infty). [\/latex]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #1<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWhat is the domain of [latex] f(x)=\\log_5(x-2)+1? [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<section>\r\n<div class=\"os-note-body\">\r\n<div id=\"ti_04_04_01\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165134274544\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 2: Identifying the Domain of a Logarithmic Shift and Reflection<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWhat is the domain of [latex] f(x)=\\log(5-2x)? [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>The logarithmic function is defined only when the input is positive, so this function is defined when [latex] 5-2x&gt;0. [\/latex] Solving this inequality,\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{lll} 5-2x &amp;&gt;&amp; 0 &amp;&amp; \\text{The input must be positive.} \\\\ -2x &amp;&gt;&amp; -5 &amp;&amp; \\text{Subtract 5.} \\\\ x &amp;&lt;&amp; \\frac{5}{2} &amp;&amp; \\text{Divide by -2 and switch the inequality.} \\end{array} [\/latex]<\/p>\r\nThe domain of [latex] f(x)=\\log(5-2x) [\/latex] is [latex] \\left(-\\infty, \\frac{5}{2}\\right). [\/latex]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/section><\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWhat is the domain of [latex] f(x)=\\log(x-5)+2? [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<section id=\"fs-id1165137554026\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Graphing Logarithmic Functions<\/h2>\r\n<p id=\"fs-id1165134104063\">Now that we have a feel for the set of values for which a logarithmic function is defined, we move on to graphing logarithmic functions. The family of logarithmic functions includes the parent function [latex] y=\\log_b(x) [\/latex] along with all its transformations: shifts, stretches, compressions, and reflections.<\/p>\r\n<p id=\"fs-id1165137679088\">We begin with the parent function [latex] y=\\log_b(x). [\/latex] Because every logarithmic function of this form is the inverse of an exponential function with the form [latex] y=b^x, [\/latex] their graphs will be reflections of each other across the line [latex] y=x. [\/latex] To illustrate this, we can observe the relationship between the input and output values of [latex] y=2^x [\/latex] and its equivalent [latex] x=\\log_2(y) [\/latex] in Table 1.<\/p>\r\n\r\n<div id=\"Table_04_04_01\" class=\"os-table\">\r\n<table class=\"grid\" data-id=\"Table_04_04_01\"><caption>Table 1<\/caption>\r\n<tbody>\r\n<tr>\r\n<td data-align=\"center\">[latex] x [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] -3 [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] -2 [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] -1 [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] 0 [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] 1 [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] 2 [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] 3 [\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">[latex] 2^x=y [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] \\frac{1}{8} [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] \\frac{1}{4} [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] \\frac{1}{2} [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] 1 [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] 2 [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] 4 [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] 8 [\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">[latex] \\log_2(y)=x [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] -3 [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] -2 [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] -1 [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] 0 [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] 1 [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] 2 [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] 3 [\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"os-caption-container\">\r\n\r\nUsing the inputs and outputs from Table 1, we can build another table to observe the relationship between points on the graphs of the inverse functions [latex] f(x)=2^x [\/latex] and [latex] g(x)=\\log_2(x). [\/latex] See Table 2.\r\n<div id=\"Table_04_04_02\" class=\"os-table\">\r\n<table class=\"grid\" data-id=\"Table_04_04_02\"><caption>Table 2<\/caption>\r\n<tbody>\r\n<tr>\r\n<td data-align=\"center\">[latex] f(x)=2^x [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] (-3, \\frac{1}{8}) [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] (-2, \\frac{1}{4}) [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] (-1, \\frac{1}{2}) [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] (0, 1) [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] (1, 2) [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] (2, 4) [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] (3, 8) [\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">[latex] g(x)=\\log_2(x) [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] (\\frac{1}{8}, -3) [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] (\\frac{1}{4}, -2) [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] (\\frac{1}{2}, -1) [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] (1, 0) [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] (2, 1) [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] (4, 2) [\/latex]<\/td>\r\n<td data-align=\"center\">[latex] (8, 3) [\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"os-caption-container\">\r\n\r\nAs we\u2019d expect, the <em data-effect=\"italics\">x<\/em>- and <em data-effect=\"italics\">y<\/em>-coordinates are reversed for the inverse functions. Figure 2 shows the graph of <em>f<\/em> and <em>g<\/em>.\r\n\r\n[caption id=\"attachment_961\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-961\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-2-300x270.webp\" alt=\"\" width=\"300\" height=\"270\" \/> Figure 4. Notice that the graphs of [latex] f(x)=2^x [\/latex] and [latex] g(x)=\\log_2(x) [\/latex] are reflections about the line [latex] y=x. [\/latex][\/caption]\r\n<p id=\"fs-id1165137406913\">Observe the following from the graph:<\/p>\r\n\r\n<ul id=\"fs-id1165137408405\">\r\n \t<li>[latex] f(x)=2^x [\/latex] has a <em data-effect=\"italics\">y<\/em>-intercept at [latex] (0, 1) [\/latex] and [latex] g(x)=\\log_2(x) [\/latex] has an <em data-effect=\"italics\">x<\/em>- intercept at [latex] (1, 0). [\/latex]<\/li>\r\n \t<li>The domain of [latex] f(x)=2^x, (-\\infty, \\infty), [\/latex] is the same as the range of [latex] g(x)=\\log_2(x). [\/latex]<\/li>\r\n \t<li>The range of [latex] f(x)=2^x, (0, \\infty), [\/latex] is the same as the domain of [latex] g(x)=\\log_2(x). [\/latex]<\/li>\r\n<\/ul>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\">Characteristics of the Graph of the Parent Function, [latex] f(x)=\\log_b(x): [\/latex]<\/span><\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFor any real number x and constant [latex] b&gt; 0, b\\not=1, [\/latex]\u00a0we can see the following characteristics in the graph of [latex] f(x)=\r\nlog_b(x): [\/latex]\r\n<ul>\r\n \t<li>one-to-one function<\/li>\r\n \t<li>vertical asymptote: [latex] x=0 [\/latex]<\/li>\r\n \t<li>domain: [latex] (0, \\infty) [\/latex]<\/li>\r\n \t<li>range: [latex] (-\\infty, \\infty) [\/latex]<\/li>\r\n \t<li><em data-effect=\"italics\">x-<\/em>intercept: [latex] (1, 0) [\/latex] and key point [latex] (b, 1) [\/latex]<\/li>\r\n \t<li><em data-effect=\"italics\">y<\/em>-intercept: none<\/li>\r\n \t<li>increasing if [latex] b&gt; 1 [\/latex]<\/li>\r\n \t<li>decreasing if [latex] 0&lt; b&lt; 1 [\/latex]<\/li>\r\n<\/ul>\r\nSee Figure 3.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_962\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-962\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-3-300x134.webp\" alt=\"\" width=\"300\" height=\"134\" \/> Figure 3[\/caption]\r\n\r\nFigure 4 shows how changing the base <em>b<\/em> in [latex] f(x)=\\log_b(x) [\/latex] can affect the graphs. Observe that the graphs compress vertically as the value of the base increases. (<em data-effect=\"italics\">Note:<\/em> recall that the function [latex] \\ln(x) [\/latex] has base [latex] e\\approx 2.718.) [\/latex]\r\n\r\n[caption id=\"attachment_963\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-963\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-4-300x224.webp\" alt=\"\" width=\"300\" height=\"224\" \/> Figure 4. The graphs of three logarithmic functions with different bases, all greater than 1.[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Given a logarithmic function with the form [latex] f(x)=\\log_b(x), [\/latex] graph the function <\/strong>\r\n<ol>\r\n \t<li>Draw and label the vertical asymptote, [latex] x=0. [\/latex]<\/li>\r\n \t<li>Plot the <em data-effect=\"italics\">x-<\/em>intercept, [latex] (1, 0). [\/latex]<\/li>\r\n \t<li>Plot the key point [latex] (b, 0). [\/latex]<\/li>\r\n \t<li>Draw a smooth curve through the points.<\/li>\r\n \t<li>State the domain, [latex] (0, \\infty), [\/latex] the range, [latex] (-\\infty, \\infty), [\/latex] and the vertical asymptote, [latex] x=0. [\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 3: Graphing a Logarithmic Function with the Form [latex] f(x)=\\log_b(x). [\/latex].<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nGraph [latex] f(x)=\\log_5(x). [\/latex] State the domain, range, and asymptote.\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Before graphing, identify the behavior and key points for the graph.\r\n<ul>\r\n \t<li>Since\u00a0[latex] b=5 [\/latex] is greater than one, we know the function is increasing. The left tail of the graph will approach the vertical asymptote [latex] x=0 [\/latex] and the right tail will increase slowly without bound.<\/li>\r\n \t<li>The <em data-effect=\"italics\">x<\/em>-intercept is [latex] (1, 0). [\/latex]<\/li>\r\n \t<li>The key point [latex] (5, 1) [\/latex] is on the graph.<\/li>\r\n \t<li>We draw and label the asymptote, plot and label the points, and draw a smooth curve through the points (see Figure 5).<\/li>\r\n<\/ul>\r\n[caption id=\"attachment_964\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-964\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-5-300x226.webp\" alt=\"\" width=\"300\" height=\"226\" \/> Figure 5[\/caption]\r\n\r\nThe domain is [latex] (0, \\infty), [\/latex] the range is [latex] (-\\infty, \\infty), [\/latex] and the vertical asymptote is [latex] x=0. [\/latex]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #3<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nGraph [latex] f(x)=\\log_{\\frac{1}{5}}(x). [\/latex] State the domain, range, and asymptote.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1165137430980\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Graphing Transformations of Logarithmic Functions<\/h2>\r\n<p id=\"fs-id1165137430986\">As we mentioned in the beginning of the section, transformations of logarithmic graphs behave similarly to those of other parent functions. We can shift, stretch, compress, and reflect the parent function [latex] y=\\log_b(x) [\/latex] without loss of shape.<\/p>\r\n\r\n<section id=\"fs-id1165137734884\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Graphing a Horizontal Shift of <em data-effect=\"italics\">f<\/em>(<em data-effect=\"italics\">x<\/em>) = log<sub><em data-effect=\"italics\">b<\/em><\/sub>(<em data-effect=\"italics\">x<\/em>)<\/h3>\r\n<p id=\"fs-id1165135530294\">When a constant <em>c<\/em> is added to the input of the parent function [latex] f(x)=\\log_b(x), [\/latex] the result is a horizontal shift <em>c<\/em> units in the <em data-effect=\"italics\">opposite<\/em> direction of the sign on <em>c<\/em>. To visualize horizontal shifts, we can observe the general graph of the parent function [latex] f(x)=\\log_b(x) [\/latex] and for [latex] c&gt; 0 [\/latex] alongside the shift left, [latex] g(x)=\\log_b(x+c), [\/latex] and the shift right, [latex] h(x)=\\log_b(x-c). [\/latex] See Figure 6.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_965\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-965\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-6-300x175.webp\" alt=\"\" width=\"300\" height=\"175\" \/> Figure 6[\/caption]\r\n\r\n<div id=\"fs-id1165135296307\" class=\"ui-has-child-title\" data-type=\"note\"><header>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\">Horizontal Shifts of the Parent Function [latex] f(x)=\\log_b(c) [\/latex]<\/span><\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFor any constant <em>c<\/em>, the function [latex] f(x)=\\log_b(x+c) [\/latex]\r\n<ul>\r\n \t<li>shifts the parent function [latex] y=\\log_b(x) [\/latex] left c units if [latex] c&gt; 0. [\/latex]<\/li>\r\n \t<li>shifts the parent function [latex] y=\\log_b(x) [\/latex] right c units if [latex] c&lt; 0. [\/latex]<\/li>\r\n \t<li>has the vertical asymptote [latex] x=-c. [\/latex]<\/li>\r\n \t<li>has domain [latex] (-c, \\infty). [\/latex]<\/li>\r\n \t<li>has range [latex] (-\\infty, \\infty). [\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<\/header><\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Given a logarithmic function with the form [latex] f(x)=\\log_b(x+c), [\/latex] graph the translation. <\/strong>\r\n<ol>\r\n \t<li>Identify the horizontal shift:\r\n<ol>\r\n \t<li>If [latex] c&gt; 0, [\/latex] shift the graph of [latex] f(x)=\\log_b(x) [\/latex] left <em>c<\/em> units.<\/li>\r\n \t<li>If [latex] c&lt; 0, [\/latex] shift the graph of [latex] f(x)=\\log_b(x) [\/latex] right <em>c<\/em> units.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Draw the vertical asymptote [latex] x=-c. [\/latex]<\/li>\r\n \t<li>Identify three key points from the parent function. Find new coordinates for the shifted functions by subtracting c from the x-coordinate.<\/li>\r\n \t<li>Label the three points.<\/li>\r\n \t<li>The Domain is [latex] (-c, \\infty), [\/latex] the range is [latex] (-\\infty, \\infty), [\/latex] and the vertical asymptote is [latex] x=-c. [\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 4:Graphing a Horizontal Shift of the Parent Function [latex] y=\\log_b(x) [\/latex]<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSketch the horizontal shift [latex] f(x)=\\log_3(x-2) [\/latex] alongside its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Since the function is [latex] f(x)=\\log_3(x-2), [\/latex] we notice x+(-2)=x-2.\r\n\r\nThus\u00a0 [latex] c=-2, [\/latex] so [latex] c&lt; 0. [\/latex] This means we will shift the function [latex]f(x)=\\log_3(x) [\/latex] right 2 units.\r\n\r\nThe vertical asymptote is [latex] x=-(-2) [\/latex] or [latex] x=2. [\/latex]\r\n\r\nConsider the three key points from the parent function, [latex] \\left(\\frac{1}{3}, -1\\right), (1, 0), [\/latex] and [latex] (3, 1). [\/latex]\r\n\r\nThe new coordinates are found by adding 2 to the <em>x<\/em>-coordinates.\r\n\r\nLabel the points [latex] \\left(\\frac{7}{3}, -1\\right), (3, 0), [\/latex] and [latex] (5, 1). [\/latex]\r\n\r\nThe domain is [latex] (2, \\infty), [\/latex] the range is [latex] (-\\infty, \\infty), [\/latex] and the vertical asymptote is [latex] x=2. [\/latex]\r\n\r\n[caption id=\"attachment_966\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-966\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-7-300x224.webp\" alt=\"\" width=\"300\" height=\"224\" \/> Figure 7[\/caption]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #4<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSketch a graph of [latex] f(x)=\\log_3(x+4) [\/latex] alongside its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1165135403538\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Graphing a Vertical Shift of <em data-effect=\"italics\">y<\/em> = log<sub><em data-effect=\"italics\">b<\/em><\/sub>(<em data-effect=\"italics\">x<\/em>)<\/h3>\r\n<p id=\"fs-id1165134310784\">When a constant <em>d<\/em> is added to the parent function [latex] f(x)=\\log_b(x), [\/latex] the result is a vertical shift <em>d<\/em> units in the direction of the sign on <em>d<\/em>. To visualize vertical shifts, we can observe the general graph of the parent function [latex] f(x)=\\log_b(x) [\/latex] alongside the shift up, [latex] g(x)=\\log_b(x)+d [\/latex] and the shift down, [latex] h(x)=\\log_b(x)-d. [\/latex] See Figure 8.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_967\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-967\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-8-300x228.webp\" alt=\"\" width=\"300\" height=\"228\" \/> Figure 8[\/caption]\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\">Vertical Shifts of the Parent Function [latex] y=\\log_b(x) [\/latex]<\/span><\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFor any constant d, the function [latex] f(x)=\\log_b(x)+d [\/latex]\r\n<ul>\r\n \t<li>shifts the parent function [latex] f(x)=\\log_b(x) [\/latex] up d units if [latex] d&gt; 0. [\/latex]<\/li>\r\n \t<li>shifts the parent function [latex] f(x)=\\log_b(x) [\/latex] down d units if [latex] d&lt; 0. [\/latex]<\/li>\r\n \t<li>has the vertical asymptote [latex] x=0. [\/latex]<\/li>\r\n \t<li>has domain [latex] (0, \\infty). [\/latex]<\/li>\r\n \t<li>has range [latex] (-\\infty, \\infty). [\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Given a logarithmic function with the form [latex] f(x)=\\log_b(x)+d, [\/latex] graph the translation. <\/strong>\r\n<ol>\r\n \t<li>Identify the vertical shift:\r\n<ol>\r\n \t<li>If [latex] d&gt; 0, [\/latex] shift the graph of [latex] f(x)=\\log_b(x) [\/latex] up d units.<\/li>\r\n \t<li>If [latex] d&lt; 0, [\/latex] shift the graph of [latex] f(x)=\\log_b(x) [\/latex] down d units.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Draw the vertical asymptote [latex] x=0. [\/latex]<\/li>\r\n \t<li>Identify three key points from the parent function. Find new coordinates for the shifted functions by adding d to the y-coordinate.<\/li>\r\n \t<li>Label the three points.<\/li>\r\n \t<li>The domain is [latex] (0, \\infty), [\/latex] the range is [latex] (-\\infty, \\infty), [\/latex] and the vertical asymptote is [latex] x=0. [\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 5: Graphing a Vertical Shift of the Parent Function [latex] y=\\log_b(x) [\/latex]<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSketch a graph of [latex] f(x)=\\log_3(x)-2 [\/latex] alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Since the function is [latex] f(x)=\\log_3(x)-2, [\/latex] we will notice [latex] d=-2. [\/latex] Thus [latex] d&lt; 0. [\/latex]\r\n\r\nThis means we will shift the function [latex] f(x)=\\log_3(2) [\/latex] down 2 units.\r\n\r\nThe vertical asymptote is [latex] x=0. [\/latex]\r\n\r\nConsider the three key points from the parent function, [latex] \\left(\\frac{1}{3}, -1\\right), (1, 0), [\/latex] and [latex] (3, 1). [\/latex]\r\n<p id=\"fs-id1165135503945\">The new coordinates are found by subtracting 2 from the <em data-effect=\"italics\">y <\/em>coordinates.<\/p>\r\nLabel the points [latex] \\left(\\frac{1}{3}, -3\\right), (1, -2), [\/latex]\u00a0and [latex] (3, -1). [\/latex]\r\n\r\n[caption id=\"attachment_968\" align=\"aligncenter\" width=\"283\"]<img class=\"size-medium wp-image-968\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-9-283x300.webp\" alt=\"\" width=\"283\" height=\"300\" \/> Figure 9[\/caption]\r\n\r\nThe domain is [latex] (0, \\infty), [\/latex] the range is [latex] (-\\infty, \\infty), [\/latex] and the vertical asymptote is [latex] x=0. [\/latex]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #5<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSketch a graph of [latex] f(x)=\\log_2(x)+2 [\/latex] alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1165137770245\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Graphing Stretches and Compressions of <em data-effect=\"italics\">y<\/em> = log<sub><em data-effect=\"italics\">b<\/em><\/sub>(<em data-effect=\"italics\">x<\/em>)<\/h3>\r\n<p id=\"fs-id1165137418005\">When the parent function [latex] f(x)=\\log_b(x) [\/latex] is multiplied by a constant [latex] a&gt; 0 [\/latex] the result is a vertical stretch or compression of the original graph. To visualize stretches and compressions, we set [latex] a&gt; 1 [\/latex] and observe the general graph of the parent function [latex] f(x)=\\log_b(x) [\/latex] alongside the vertical stretch, [latex] g(x)=a\\log_b(x) [\/latex] and the vertical compression, [latex] h(x)=\\frac{1}{a}\\log_b(x). [\/latex] See Figure 10.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_969\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-969\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-10-300x233.webp\" alt=\"\" width=\"300\" height=\"233\" \/> Figure 10[\/caption]\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\">Vertical Stretches and Compressions of the Parent Function- [latex] y=\\log_b(x) [\/latex]<\/span><\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFor any constant [latex] a&gt; 1, [\/latex] the function [latex] f(x)=a\\log_b(x) [\/latex]\r\n<ul>\r\n \t<li>stretches the parent function [latex] y=\\log_b(x) [\/latex] vertically by a factor of a if [latex] a&gt; 1. [\/latex]<\/li>\r\n \t<li>compresses the parent function [latex] y=\\log_b(x) [\/latex] vertically by a factor of a if [latex] 0&lt; a&lt; 1. [\/latex]<\/li>\r\n \t<li>has the vertical asymptote [latex] x=0. [\/latex]<\/li>\r\n \t<li>has the <em data-effect=\"italics\">x<\/em>-intercept [latex] (1, 0). [\/latex]<\/li>\r\n \t<li>has domain [latex] (0, \\infty). [\/latex]<\/li>\r\n \t<li>has range [latex] (-\\infty, \\infty). [\/latex]<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Given a logarithmic function with the form [latex] f(x)=a\\log_b(x), a&gt; 0, [\/latex] graph the translation. <\/strong>\r\n<ol>\r\n \t<li>Identify the vertical stretch or compressions:\r\n<ol>\r\n \t<li>If [latex] |a|&gt; 1, [\/latex] the graph of [latex] f(x)=\\log_b(x) [\/latex] is stretched by a factor of a units.<\/li>\r\n \t<li>If [latex] |a|&lt; 1, [\/latex] the graph of [latex] f(x)=\\log_b(x) [\/latex] is compressed by a factor of a units.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li>Draw the vertical asymptote [latex] x=0. [\/latex]<\/li>\r\n \t<li>Identify three key points from the parent function. Find new coordinates for the shifted functions by multiplying the <em>y<\/em>-coordinates by <em>a<\/em>.<\/li>\r\n \t<li>Label the three points.<\/li>\r\n \t<li>The domain is [latex] (0, \\infty), [\/latex] the range is [latex] (-\\infty, \\infty), [\/latex] and the vertical asymptote is [latex] x=0. [\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 6: Graphing a Stretch or Compression of the Parent Function [latex] y=\\log_b(x) [\/latex]<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSketch a graph of [latex] f(x)=2\\log_4(x) [\/latex] alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Since the function is [latex] f(x)=2\\log_4(x), [\/latex] we will notice [latex] a=2. [\/latex]\r\n\r\nThis means we will stretch the function [latex] f(x)=\\log_4(x) [\/latex] by a factor of 2.\r\n\r\nThe vertical asymptote is [latex] x=0. [\/latex]\r\n\r\nConsider the three key points from the parent function, [latex] \\left(\\frac{1}{4}, -1\\right), (1, 0), [\/latex] and [latex] (4, 1). [\/latex]\r\n<p id=\"fs-id1165135570058\">The new coordinates are found by multiplying the <em>y<\/em>-coordinates by 2.<\/p>\r\nLabel the points [latex] \\left(\\frac{1}{4}, -2\\right), (1, 0), [\/latex] and [latex] (4, 2). [\/latex]\r\n\r\nThe domain is [latex] (0, \\infty), [\/latex] the range is [latex] (-\\infty, \\infty), [\/latex] and the vertical asymptote is [latex] x=0. [\/latex] See Figure 11.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_970\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-970\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-11-300x225.webp\" alt=\"\" width=\"300\" height=\"225\" \/> Figure 11[\/caption]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #6<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSketch a graph of [latex] f(x)=\\frac{1}{2}\\log_4(x) [\/latex] alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 7: Combining a Shift and a Stretch<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSketch a graph of [latex] f(x)=5\\log(x+2). [\/latex] State the domain, range, and asymptote.\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Remember: what happens inside parentheses happens first. First, we move the graph left 2 units, then stretch the function vertically by a factor of 5, as in Figure 12. The vertical asymptote will be shifted to [latex] x=-2. [\/latex] The <em data-effect=\"italics\">x<\/em>-intercept will be [latex] (-1, 0). [\/latex] The domain will be [latex] (-2, \\infty). [\/latex] Two points will help give the shape of the graph: [latex] (-1, 0) [\/latex] and [latex] (8, 5). [\/latex] We chose [latex] x=8 [\/latex] as the <em data-effect=\"italics\">x<\/em>-coordinate of one point to graph because when [latex] x=8, x+2=10, [\/latex]\u00a0the base of the common logarithm.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_971\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-971\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-12-300x272.webp\" alt=\"\" width=\"300\" height=\"272\" \/> Figure 12[\/caption]\r\n\r\nThe domain is [latex] (-2, \\infty) [\/latex] the range is [latex] (-\\infty, \\infty), [\/latex] and the vertical asymptote is [latex] x=-2. [\/latex]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #7<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSketch a graph of the function [latex] f(x)=3\\log(x-2)+1. [\/latex] State the domain, range, and asymptote.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1165137629003\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Graphing Reflections of [latex] f(x)=\\log_b(x) [\/latex]<\/h3>\r\n<p id=\"fs-id1165135169315\">When the parent function [latex] f(x)=\\log_b(x) [\/latex] is multiplied by -1, the result is a reflection about the <em data-effect=\"italics\">x<\/em>-axis. When the <em data-effect=\"italics\">input<\/em> is multiplied by -1, the result is a reflection about the <em data-effect=\"italics\">y<\/em>-axis. To visualize reflections, we restrict [latex] b&gt; 1, [\/latex] and observe the general graph of the parent function [latex] f(x)=\\log_b(x) [\/latex] alongside the reflection about the <em data-effect=\"italics\">x<\/em>-axis, [latex] g(x)=-\\log_b(x) [\/latex] and the reflection about the <em data-effect=\"italics\">y<\/em>-axis, [latex] h(x)=log_b(-x). [\/latex]<\/p>\r\n\r\n\r\n[caption id=\"attachment_972\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-972\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-13-300x242.webp\" alt=\"\" width=\"300\" height=\"242\" \/> Figure 13[\/caption]\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\">Reflections of the Parent Function [latex] y=\\log_b(x) [\/latex]<\/span><\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nThe function [latex] f(x)=-\\log_b(x) [\/latex]\r\n<ul>\r\n \t<li>reflects the parent function [latex] y=\\log_b(x) [\/latex] about the <em data-effect=\"italics\">x<\/em>-axis.<\/li>\r\n \t<li>has domain, [latex] (0, \\infty), [\/latex] range, [latex] (-\\infty, \\infty), [\/latex] and vertical asymptote, [latex] x=0, [\/latex] which are unchanged from the parent function.<\/li>\r\n<\/ul>\r\nThe function [latex] f(x)=\\log_b(-x) [\/latex]\r\n<ul>\r\n \t<li>reflects the parent function [latex] y=\\log_b(x) [\/latex] about the <em data-effect=\"italics\">y<\/em>-axis.<\/li>\r\n \t<li>has domain [latex] (-\\infty, 0). [\/latex]<\/li>\r\n \t<li>has range, [latex] (-\\infty, \\infty), [\/latex] and vertical asymptote, [latex] x=0, [\/latex] which are unchanged from the parent function.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Given a logarithmic function with the parent function [latex] f(x)=\\log_b(x), [\/latex] graph a translation. <\/strong>\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%; height: 160px;\" border=\"0\"><caption>Table 3<\/caption>\r\n<tbody>\r\n<tr style=\"height: 20px;\">\r\n<td style=\"width: 50%; height: 20px;\">If [latex] f(x)=-\\log_b(x) [\/latex]<\/td>\r\n<td style=\"width: 50%; height: 20px;\">If [latex] f(x)=\\log_b(-x) [\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 47px;\">\r\n<td style=\"width: 50%; height: 47px;\">1. Draw the vertical asymptote, [latex] x=0. [\/latex]<\/td>\r\n<td style=\"width: 50%; height: 47px;\">1. Draw the vertical asymptote, [latex] x=0. [\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 48px;\">\r\n<td style=\"width: 50%; height: 48px;\">2. Plot the <em data-effect=\"italics\">x-<\/em>intercept, [latex] (1, 0). [\/latex]<\/td>\r\n<td style=\"width: 50%; height: 48px;\">2. Plot the <em data-effect=\"italics\">x-<\/em>intercept, [latex] (1, 0). [\/latex]<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 50%; height: 15px;\">3. Reflect the graph of the parent function [latex] f(x)=\\log_b(x) [\/latex] about the <em data-effect=\"italics\">x<\/em>-axis.<\/td>\r\n<td style=\"width: 50%; height: 15px;\">3. Reflect the graph of the parent function [latex] f(x)=\\log_b(x) [\/latex] about the <em data-effect=\"italics\">y<\/em>-axis.<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 50%;\" data-align=\"left\">4. Draw a smooth curve through the points.<\/td>\r\n<td style=\"width: 50%; height: 15px;\">4. Draw a smooth curve through the points.<\/td>\r\n<\/tr>\r\n<tr style=\"height: 15px;\">\r\n<td style=\"width: 50%; height: 15px;\">5. State the domain, [latex] (0, \\infty), [\/latex] the range, [latex] (-\\infty, \\infty), [\/latex] and the vertical asymptote [latex] x=0. [\/latex]<\/td>\r\n<td style=\"width: 50%; height: 15px;\">5. State the domain, [latex] (-\\infty, 0), [\/latex] the range, [latex] (-\\infty, \\infty), [\/latex] and the vertical asymptote [latex] x=0. [\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 8: Graphing a Reflection of a Logarithmic Function<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSketch a graph of [latex] f(x)=\\log(-x) [\/latex] alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Before graphing [latex] f(x)=\\log(-x), [\/latex] identify the behavior and key points for the graph.\r\n<ul>\r\n \t<li>Since [latex] b=10 [\/latex] is greater than one, we know that the parent function is increasing. Since the <em data-effect=\"italics\">input<\/em> value is multiplied by -1, <em>f<\/em> is a reflection of the parent graph about the <em data-effect=\"italics\">y-<\/em>axis. Thus, [latex] f(x)=\\log(-x) [\/latex] will be decreasing as <em>x<\/em> moves from negative infinity to zero, and the right tail of the graph will approach the vertical asymptote [latex] x=0. [\/latex]<\/li>\r\n \t<li>The <em data-effect=\"italics\">x<\/em>-intercept is [latex] (-1, 0). [\/latex]<\/li>\r\n \t<li>We draw and label the asymptote, plot and label the points, and draw a smooth curve through the points.<\/li>\r\n<\/ul>\r\n[caption id=\"attachment_973\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-973\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-14-300x223.webp\" alt=\"\" width=\"300\" height=\"223\" \/> Figure 14[\/caption]\r\n\r\nThe domain is [latex] (-\\infty, 0), [\/latex] the range is [latex] (-\\infty, \\infty), [\/latex] and the vertical asymptote is [latex] x=0. [\/latex]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #8<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nGraph [latex] f(x)=-\\log(-x). [\/latex] State the domain, range, and asymptote.\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">How To<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Given a logarithmic equation, use a graphing calculator to approximate solutions.<\/strong>\r\n<ol>\r\n \t<li>Press <strong>[Y=]<\/strong>. Enter the given logarithm equation or equations as <strong>Y<sub>1<\/sub>=<\/strong> and, if needed, <strong>Y<sub>2<\/sub>=<\/strong>.<\/li>\r\n \t<li>Press <strong>[GRAPH]<\/strong> to observe the graphs of the curves and use <strong>[WINDOW]<\/strong> to find an appropriate view of the graphs, including their point(s) of intersection.<\/li>\r\n \t<li>To find the value of <em>x<\/em>, we compute the point of intersection. Press <strong>[2ND] <\/strong>then <strong>[CALC]<\/strong>. Select \u201cintersect\u201d and press <strong>[ENTER]<\/strong> three times. The point of intersection gives the value of <em>x<\/em> for the point(s) of intersection.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 9: Approximating the Solution of a Logarithmic Equation<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSolve [latex] 4\\ln(x)+1=-2\\ln(x-1) [\/latex] graphically. Round to the nearest thousandth.\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>Press <strong>[Y=]<\/strong> and enter [latex] 4\\ln(x)+1 [\/latex] next to <strong>Y<sub>1<\/sub><\/strong>=. Then enter [latex] -2\\ln(x-1) [\/latex] next to <strong>Y<sub>2<\/sub>=<\/strong>. For a window, use the values 0 to 5 for <em>x<\/em> and \u201310 to 10 for <em>y<\/em>. Press <strong>[GRAPH]<\/strong>. The graphs should intersect somewhere a little to right of [latex] x=1. [\/latex]\r\n\r\nFor a better approximation, press <strong>[2ND] <\/strong>then <strong>[CALC]<\/strong>. Select <strong>[5: intersect]<\/strong> and press <strong>[ENTER]<\/strong> three times. The <em data-effect=\"italics\">x<\/em>-coordinate of the point of intersection is displayed as 1.3385297. (Your answer may be different if you use a different window or use a different value for <strong>Guess?<\/strong>) So, to the nearest thousandth, [latex] x\\approx 1.339. [\/latex]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #9<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSolve [latex] 5\\log(x+2)=4-\\log(x) [\/latex] graphically. Round to the nearest thousandth.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1165135528930\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Summarizing Translations of the Logarithmic Function<\/h3>\r\n<p id=\"fs-id1165135528935\">Now that we have worked with each type of translation for the logarithmic function, we can summarize each in Table 4 to arrive at the general equation for translating exponential functions.<\/p>\r\n\r\n<div id=\"Table_04_04_009\" class=\"os-table\">\r\n<table class=\"grid\" data-id=\"Table_04_04_009\"><caption>Table 4<\/caption>\r\n<thead>\r\n<tr>\r\n<th colspan=\"2\" scope=\"colgroup\">Transformations of the Parent Function [latex] y=\\log_b(x) [\/latex]<\/th>\r\n<\/tr>\r\n<tr>\r\n<th scope=\"col\" data-align=\"center\">Transformation<\/th>\r\n<th scope=\"col\" data-align=\"center\">Form<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td data-align=\"center\">Shift\r\n<ul id=\"fs-id1165137416971\">\r\n \t<li>Horizontally <em>c<\/em> units to the left<\/li>\r\n \t<li>Vertically <em>d<\/em> units up<\/li>\r\n<\/ul>\r\n<\/td>\r\n<td data-align=\"center\">[latex] y=\\log_b(x+c)+d [\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">Stretch and Compress\r\n<ul id=\"fs-id1165137427553\">\r\n \t<li>Stretch if [latex] |a|&gt; 1 [\/latex]<\/li>\r\n \t<li>Compression if [latex] |a|&lt; 1 [\/latex]<\/li>\r\n<\/ul>\r\n<\/td>\r\n<td data-align=\"center\">[latex] y=a\\log_b(x) [\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">Reflect about the <em data-effect=\"italics\">x<\/em>-axis<\/td>\r\n<td data-align=\"center\">[latex] y=-\\log_b(x) [\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">Reflect about the <em data-effect=\"italics\">y<\/em>-axis<\/td>\r\n<td data-align=\"center\">[latex] y-\\log_b(-x) [\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td data-align=\"center\">General equation for all translations<\/td>\r\n<td data-align=\"center\">[latex] y=a\\log_b(x+c)+d [\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<div class=\"os-caption-container\">\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Translations of Logarithmic Functions<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nAll translations of the parent logarithmic function, [latex] y=\\log_b(x), [\/latex] have the form\r\n<p style=\"text-align: center;\">[latex] f(x)=a\\log_b(x+c)+d [\/latex]<\/p>\r\nwhere the parent function, [latex] y=\\log_b(x), b&gt; 1, [\/latex] is\r\n<ul>\r\n \t<li>shifted vertically up <em>d<\/em> units.<\/li>\r\n \t<li>shifted horizontally to the left <em>c<\/em> units.<\/li>\r\n \t<li>stretched vertically by a factor of [latex] |a| [\/latex] if [latex] |a|&gt; 0. [\/latex]<\/li>\r\n \t<li>compressed vertically by a factor of [latex] |a| [\/latex] if [latex] 0&lt; |a|&lt; 1. [\/latex]<\/li>\r\n \t<li>reflected about the <em data-effect=\"italics\">x-<\/em>axis when [latex] a&lt; 0. [\/latex]<\/li>\r\n<\/ul>\r\nFor [latex] f(x)=\\log(-x), [\/latex] the graph of the parent function is reflected about the <em data-effect=\"italics\">y<\/em>-axis.\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 10: Finding the Vertical Asymptote of a Logarithm Graph<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWhat is the vertical asymptote of [latex] f(x)=-2\\log_3(x+4)+5? [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>The vertical asymptote is at [latex] x=-4. [\/latex]\r\n<h3>Analysis<\/h3>\r\nThe coefficient, the base, and the upward translation do not affect the asymptote. The shift of the curve 4 units to the left shifts the vertical asymptote to [latex] x=-4. [\/latex]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #10<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWhat is the vertical asymptote of [latex] f(x)=3+\\ln(x-1)? [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<section>\r\n<div class=\"os-note-body\">\r\n<div id=\"ti_04_04_10\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\"><section>\r\n<div id=\"fs-id1165137748692\" data-type=\"problem\">\r\n<div class=\"os-problem-container\">\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 11: Finding the Equation from a Graph<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nFind a possible equation for the common logarithmic function graphed in Figure 15.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_974\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-974\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-15-300x226.webp\" alt=\"\" width=\"300\" height=\"226\" \/> Figure 15[\/caption]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>This graph has a vertical asymptote at [latex] x=-2 [\/latex] and has been vertically reflected. We do not know yet the vertical shift or the vertical stretch. We know so far that the equation will have form:\r\n<p style=\"text-align: center;\">[latex] f(x)=-a\\log(x_2)+k [\/latex]<\/p>\r\nIt appears the graph passes through the points [latex] (-1, 1) [\/latex] and [latex] (2, -1). [\/latex] Substituting [latex] (-1, 1), [\/latex]\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{lll} 1&amp;=&amp; -a\\log(-1+2)+k &amp;&amp; \\text{Substitute } (-1, 1). \\\\ 1 &amp;=&amp; -a\\log(1)+k &amp;&amp; \\text{Arithmetic.} \\\\ 1 &amp;=&amp; k &amp;&amp; \\log(1)=0. \\end{array} [\/latex]<\/p>\r\nNext, substituting in [latex] (2, -1), [\/latex]\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{lll} -1 &amp;=&amp; -a\\log(2+2)+1 &amp;&amp; \\text{Plug in } (2, -1). \\\\ -2 &amp;=&amp; -a\\log(4) &amp;&amp; \\text{Arithmetic.} \\\\ a &amp;=&amp; \\frac{2}{\\log(4)} &amp;&amp; \\text{Solve for } a. \\end{array} [\/latex]<\/p>\r\nThis gives us the equation [latex] f(x)=-\\frac{2}{\\log(4)}\\log(x+2)+1. [\/latex]\r\n<h3>Analysis<\/h3>\r\nWe can verify this answer by comparing the function values in Table 5 with the points on the graph in Figure 15.\r\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\" border=\"0\"><caption>Table 5<\/caption>\r\n<tbody>\r\n<tr>\r\n<td style=\"width: 16.6667%;\">[latex] x [\/latex]<\/td>\r\n<td style=\"width: 16.6667%;\">-1<\/td>\r\n<td style=\"width: 16.6667%;\">0<\/td>\r\n<td style=\"width: 16.6667%;\">1<\/td>\r\n<td style=\"width: 16.6667%;\">2<\/td>\r\n<td style=\"width: 16.6667%;\">3<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 16.6667%;\">[latex] f(x) [\/latex]<\/td>\r\n<td style=\"width: 16.6667%;\">1<\/td>\r\n<td style=\"width: 16.6667%;\">0<\/td>\r\n<td style=\"width: 16.6667%;\">-0\/58496<\/td>\r\n<td style=\"width: 16.6667%;\">-1<\/td>\r\n<td style=\"width: 16.6667%;\">-1.3219<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 16.6667%;\">[latex] x [\/latex]<\/td>\r\n<td style=\"width: 16.6667%;\">4<\/td>\r\n<td style=\"width: 16.6667%;\">5<\/td>\r\n<td style=\"width: 16.6667%;\">6<\/td>\r\n<td style=\"width: 16.6667%;\">7<\/td>\r\n<td style=\"width: 16.6667%;\">8<\/td>\r\n<\/tr>\r\n<tr>\r\n<td style=\"width: 16.6667%;\">[latex] f(x) [\/latex]<\/td>\r\n<td style=\"width: 16.6667%;\">-1.5850<\/td>\r\n<td style=\"width: 16.6667%;\">-1.8074<\/td>\r\n<td style=\"width: 16.6667%;\">-2<\/td>\r\n<td style=\"width: 16.6667%;\">-2.1699<\/td>\r\n<td style=\"width: 16.6667%;\">-2.3219<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/details><\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Try It #11<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nGive the equation of the natural logarithm graphed in Figure 16.\r\n\r\n&nbsp;\r\n\r\n[caption id=\"attachment_975\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-975\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-16-300x272.webp\" alt=\"\" width=\"300\" height=\"272\" \/> Figure 16[\/caption]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Q&amp;A<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Q: Is it possible to tell the domain and range and describe the end behavior of a function just by looking at the graph?<\/strong>\r\n\r\n<em data-effect=\"italics\">A: Yes, if we know the function is a general logarithmic function. For example, look at the graph in Figure 16. The graph approaches <\/em> [latex] x=-3 [\/latex] (or thereabouts) more and more closely, so [latex] x=-3 [\/latex] is, or is very close to, the vertical asymptote. It approaches from the right, so the domain is all points to the right, [latex] \\{x \\mid x &gt; -3\\}. [\/latex] The range, as with all general logarithmic functions, is all real numbers. And we can see the end behavior because the graph goes down as it goes left and up as it goes right. The end behavior is that as [latex] x\\rightarrow -3^+, f(x)\\rightarrow -\\infty [\/latex] and as [latex] x\\rightarrow \\infty, f(x)\\rightarrow \\infty. [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/section>\r\n<div class=\"textbox textbox--key-takeaways\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Media<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nAccess these online resources for additional instruction and practice with graphing logarithms.\r\n<ul>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=w1A2ZYmfGco\">Graph an Exponential Function and Logarithmic Function<\/a><\/li>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=GnfclmCE9rE\">Match Graphs with Exponential and Logarithmic Functions<\/a><\/li>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=_Om0ZMzIgUk\">Find the Domain of Logarithmic Functions<\/a><\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n<\/section>\r\n<div class=\"os-eos os-section-exercises-container\" data-uuid-key=\".section-exercises\">\r\n<h2 data-type=\"document-title\" data-rex-keep=\"true\"><span class=\"os-text\">6.4 Section Exercises<\/span><\/h2>\r\n<section id=\"fs-id1165135189917\" class=\"section-exercises\" data-depth=\"1\"><section id=\"fs-id1165135189921\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Verbal<\/h3>\r\n<div id=\"fs-id1165135582182\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135582184\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135582182-solution\">1<\/a><span class=\"os-divider\">. <\/span>The inverse of every logarithmic function is an exponential function and vice-versa. What does this tell us about the relationship between the coordinates of the points on the graphs of each?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137855282\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137855284\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span>What type(s) of translation(s), if any, affect the range of a logarithmic function?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137855292\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137855294\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137855292-solution\">3<\/a><span class=\"os-divider\">. <\/span>What type(s) of translation(s), if any, affect the domain of a logarithmic function?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137424692\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137424694\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">4<\/span><span class=\"os-divider\">. <\/span>Consider the general logarithmic function [latex] f(x)=\\log_b(x). [\/latex] Why can\u2019t <em>x<\/em> be zero?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137697126\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137697128\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137697126-solution\">5<\/a><span class=\"os-divider\">. <\/span>Does the graph of a general logarithmic function have a horizontal asymptote? Explain.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><section id=\"fs-id1165137459922\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Algebraic<\/h3>\r\n<p id=\"fs-id1165137459927\">For the following exercises, state the domain and range of the function.<\/p>\r\n\r\n<div id=\"fs-id1165137459930\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137737591\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">6<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=\\log_3(x+4) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/section><\/div>\r\n<div id=\"fs-id1165135264771\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135264773\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135264771-solution\">7<\/a><span class=\"os-divider\">. <\/span> [latex] h(x)=\\ln\\left(\\frac{1}{2}-x\\right) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135253806\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135253808\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">8<\/span><span class=\"os-divider\">. <\/span> [latex] g(x)=\\log_5(2x+9)-2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135415813\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135415815\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135415813-solution\">9<\/a><span class=\"os-divider\">. <\/span> [latex] h(x)=\\ln(4x+17)-5 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137843712\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137843714\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">10<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=\\log_2(12-3x)-3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165135503696\">For the following exercises, state the domain and the vertical asymptote of the function.<\/p>\r\n\r\n<div id=\"fs-id1165134037565\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134037567\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165134037565-solution\">11<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=\\log_b(x-5) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137933908\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137933910\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">12<\/span><span class=\"os-divider\">. <\/span> [latex] g(x)=\\ln(3-x) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137737027\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137737029\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137737027-solution\">13<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=\\log(3x+1) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135188081\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135188083\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">14<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=3\\log(-x)+2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134188310\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134188313\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165134188310-solution\">15<\/a><span class=\"os-divider\">. <\/span> [latex] g(x)=-\\ln(3x+9)-7 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165135194730\">For the following exercises, state the domain, vertical asymptote, and end behavior of the function.<\/p>\r\n\r\n<div id=\"fs-id1165135194734\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135194736\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">16<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=\\ln(2-x) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137427662\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137427665\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137427662-solution\">17<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=\\log\\left(x-\\frac{3}{7}\\right) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137431240\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137431242\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">18<\/span><span class=\"os-divider\">. <\/span> [latex] h(x)=-\\log(3x-4)+3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134194950\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134194952\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165134194950-solution\">19<\/a><span class=\"os-divider\">. <\/span> [latex] g(x)=\\ln(2x+6)-5 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135532457\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135532459\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">20<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=\\log_3(15-5x)+6 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165137426354\">For the following exercises, state the domain, range, and <em data-effect=\"italics\">x<\/em>- and <em data-effect=\"italics\">y<\/em>-intercepts, if they exist. If they do not exist, write DNE.<\/p>\r\n\r\n<div id=\"fs-id1165137426368\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137426370\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137426368-solution\">21<\/a><span class=\"os-divider\">. <\/span> [latex] h(x)=\\log_4(x-1)+1 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137755640\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135543374\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">22<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=\\log(5x+10)+3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135210139\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135210141\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135210139-solution\">23<\/a><span class=\"os-divider\">. <\/span> [latex] g(x)=\\ln(-x)-2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137810010\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137810012\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">24<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=\\log_2(x+2)-5 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137942453\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137942455\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137942453-solution\">25<\/a><span class=\"os-divider\">. <\/span> [latex] h(x)=3\\ln(x)=9 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<section id=\"fs-id1165137422264\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Graphical<\/h3>\r\n<p id=\"fs-id1165137422269\">For the following exercises, match each function in Figure 17 with the letter corresponding to its graph.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_976\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-976\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-17-300x271.webp\" alt=\"\" width=\"300\" height=\"271\" \/> Figure 17[\/caption]\r\n\r\n<div id=\"fs-id1165134032273\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134032275\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">26<\/span><span class=\"os-divider\">. <\/span> [latex] d(x)=\\log(x) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section>\r\n<div id=\"fs-id1165135194197\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137911558\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135194197-solution\">27<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=\\ln(x) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134255566\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134255568\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">28<\/span><span class=\"os-divider\">. <\/span> [latex] g(x)=\\log_2(x) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137758154\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137758156\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137758154-solution\">29<\/a><span class=\"os-divider\">. <\/span> [latex] h(x)=\\log_5(x) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135571665\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135571667\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">30<\/span><span class=\"os-divider\">. <\/span> [latex] j(h)=\\log_{25}(x) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"eip-312\">For the following exercises, match each function in Figure 18 with the letter corresponding to its graph.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_977\" align=\"aligncenter\" width=\"233\"]<img class=\"size-medium wp-image-977\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-18-233x300.webp\" alt=\"\" width=\"233\" height=\"300\" \/> Figure 18[\/caption]\r\n\r\n<div id=\"fs-id1165134380389\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135400154\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165134380389-solution\">31<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=\\log_{\\frac{1}{3}}(x) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135206089\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135206092\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">32<\/span><span class=\"os-divider\">. <\/span> [latex] g(x)=\\log_2(x) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137855381\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137855383\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137855381-solution\">33<\/a><span class=\"os-divider\">. <\/span> [latex] h(x)=\\log_{\\frac{3}{4}}(x) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"eip-967\">For the following exercises, sketch the graphs of each pair of functions on the same axis.<\/p>\r\n\r\n<div id=\"fs-id1165135394346\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135394348\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">34<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=\\log(x) \\ \\text{and } g(x)=10^x [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137862457\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137862459\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137862457-solution\">35<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=\\log(x) \\ \\text{and } g(x)=\\log_{\\frac{1}{2}}(x) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135180056\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135180058\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">36<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=\\log_4(x) \\ \\text{and } g(x)=\\ln(x) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135536589\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134325846\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135536589-solution\">37<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=e^x \\ \\text{and } g(x)=\\ln(x) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165135496626\">For the following exercises, match each function in Figure 19 with the letter corresponding to its graph.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_978\" align=\"aligncenter\" width=\"298\"]<img class=\"size-medium wp-image-978\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-19-298x300.webp\" alt=\"\" width=\"298\" height=\"300\" \/> Figure 19[\/caption]\r\n\r\n<div id=\"fs-id1165135189876\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135189878\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">38<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=\\log_4(-x+2) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137656835\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137656837\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137656835-solution\">39<\/a><span class=\"os-divider\">. <\/span> [latex] g(x)=-\\log_4(x+2) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135632110\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135632112\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">40<\/span><span class=\"os-divider\">. <\/span> [latex] h(x)=\\log_4(x+2) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165135571794\">For the following exercises, sketch the graph of the indicated function.<\/p>\r\n\r\n<div id=\"fs-id1165135571798\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135571800\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135571798-solution\">41<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=\\log_2(x+2) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137849072\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137849074\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">42<\/span><span class=\"os-divider\">. <\/span> [latex] f(x)=2\\log(x) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135251375\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135251377\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135251375-solution\">43<\/a><span class=\"os-divider\">. <\/span> [latex] f(x)=\\ln(-x) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134032422\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134032425\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">44<\/span><span class=\"os-divider\">. <\/span> [latex] g(x)=\\log(4x+16)+4 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137932668\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137932670\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137932668-solution\">45<\/a><span class=\"os-divider\">. <\/span> [latex] g(x)=\\log(6-3x)+1 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137705417\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137705419\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">46<\/span><span class=\"os-divider\">. <\/span> [latex] h(x)=-\\frac{1}{2}\\ln(x+1)-3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165135439867\">For the following exercises, write a logarithmic equation corresponding to the graph shown.<\/p>\r\n\r\n<div id=\"fs-id1165135439871\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135439873\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135439871-solution\">47<\/a><span class=\"os-divider\">. <\/span>Use [latex] y=\\log_2(x) [\/latex] as the parent function.\r\n<div class=\"os-problem-container\">\r\n\r\n<span id=\"fs-id1165135443953\" data-type=\"media\" data-alt=\"The graph y=log_2(x) has been reflected over the y-axis and shifted to the right by 1.\" data-display=\"block\">\r\n<img class=\"alignnone size-medium wp-image-979\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4.47-298x300.webp\" alt=\"\" width=\"298\" height=\"300\" \/>\r\n<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135191066\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135191068\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">48<\/span><span class=\"os-divider\">. <\/span>Use [latex] f(x)=\\log_3(x) [\/latex] as the parent function.\r\n<div class=\"os-problem-container\">\r\n\r\n<span id=\"fs-id1165137674237\" data-type=\"media\" data-alt=\"The graph y=log_3(x) has been reflected over the x-axis, vertically stretched by 3, and shifted to the left by 4.\" data-display=\"block\">\r\n<img class=\"alignnone size-medium wp-image-980\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4.48-298x300.webp\" alt=\"\" width=\"298\" height=\"300\" \/>\r\n<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137674252\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137674255\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137674252-solution\">49<\/a><span class=\"os-divider\">. <\/span>Use [latex] f(x)=\\log_4(x) [\/latex] as the parent function.\r\n<div class=\"os-problem-container\">\r\n\r\n<span id=\"fs-id1165134086070\" data-type=\"media\" data-alt=\"The graph y=log_4(x) has been vertically stretched by 3, and shifted to the left by 2.\" data-display=\"block\">\r\n<img class=\"alignnone size-medium wp-image-981\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4.49-298x300.webp\" alt=\"\" width=\"298\" height=\"300\" \/>\r\n<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135203463\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135203465\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">50<\/span><span class=\"os-divider\">. <\/span>Use [latex] f(x)=\\log_5(x) [\/latex] as the parent function.\r\n<div class=\"os-problem-container\">\r\n\r\n<span id=\"fs-id1165137705189\" data-type=\"media\" data-alt=\"The graph y=log_3(x) has been reflected over the x-axis and y-axis, vertically stretched by 2, and shifted to the right by 5.\" data-display=\"block\">\r\n<img class=\"alignnone size-medium wp-image-982\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4.50-298x300.webp\" alt=\"\" width=\"298\" height=\"300\" \/>\r\n<\/span>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<section id=\"fs-id1165137705205\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Technology<\/h3>\r\n<p id=\"fs-id1165137705211\">For the following exercises, use a graphing calculator to find approximate solutions to each equation.<\/p>\r\n\r\n<div id=\"fs-id1165137705215\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137705217\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137705215-solution\">51<\/a><span class=\"os-divider\">. <\/span> [latex] \\log(x-1)+2=\\ln(x-1)+2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section>\r\n<div id=\"fs-id1165137674311\" class=\"material-set-2\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137674313\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">52<\/span><span class=\"os-divider\">. <\/span> [latex] \\log(2x-3)+2=-\\log(2x-3)+5 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135176226\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135176229\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135176226-solution\">53<\/a><span class=\"os-divider\">. <\/span> [latex] \\ln(x-2)=-\\ln(x+1) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135181791\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135181793\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">54<\/span><span class=\"os-divider\">. <\/span> [latex] 2\\ln(5x+1)=\\frac{1}{2}\\ln(-5x)+1 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135367864\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135367866\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135367864-solution\">55<\/a><span class=\"os-divider\">. <\/span> [latex] \\frac{1}{3}\\log(1-x)=\\log(x+1)+\\frac{1}{3} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<section id=\"fs-id1165135485736\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Extensions<\/h3>\r\n<div id=\"fs-id1165135485742\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135485744\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">56<\/span><span class=\"os-divider\">. <\/span>Let <em>b<\/em> be any positive real number such that [latex] b\\not=1. [\/latex] What must [latex] \\log_b1 [\/latex] be equal to? Verify the result.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135193274\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135193276\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135193274-solution\">57<\/a><span class=\"os-divider\">. <\/span>Explore and discuss the graphs of [latex] f(x)=\\log_{\\frac{1}{2}}(x) [\/latex] and [latex] g(x)=-\\log_2(x). [\/latex] Make a conjecture based on the result.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135369592\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135369594\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">58<\/span><span class=\"os-divider\">. <\/span>Prove the conjecture made in the previous exercise.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135369601\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135369603\" data-type=\"problem\">\r\n\r\n<a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135369601-solution\">59<\/a><span class=\"os-divider\">. <\/span>What is the domain of the function [latex] f(x)=\\ln\\left(\\frac{x+2}{x-4}\\right)? [\/latex] Discuss the result.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137838658\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137838660\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">60<\/span><span class=\"os-divider\">. <\/span>Use properties of exponents to find the <em data-effect=\"italics\">x<\/em>-intercepts of the function [latex] f(x)=\\log(x^2+4x+4) [\/latex] algebraically. Show the steps for solving, and then verify the result by graphing the function.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section>","rendered":"<div id=\"main-content\" class=\"MainContent__ContentStyles-sc-6yy1if-0 NnXKu\" tabindex=\"-1\" data-dynamic-style=\"true\">\n<div id=\"page_44418435-ed46-454a-aba4-cd57f5266654\" class=\"chapter-content-module\" data-type=\"page\" data-book-content=\"true\">\n<div class=\"textbox textbox--learning-objectives\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Learning Objectives<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>In this section, you will:<\/p>\n<ul>\n<li>Identify the domain of a logarithmic function.<\/li>\n<li>Graph logarithmic functions.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/div>\n<p id=\"fs-id1165135194555\">In Graphs of Exponential Functions, we saw how creating a graphical representation of an exponential model gives us another layer of insight for predicting future events. How do logarithmic graphs give us insight into situations? Because every logarithmic function is the inverse function of an exponential function, we can think of every output on a logarithmic graph as the input for the corresponding inverse exponential equation. In other words, logarithms give the <em data-effect=\"italics\">cause<\/em> for an <em data-effect=\"italics\">effect<\/em>.<\/p>\n<p id=\"fs-id1165137603580\">To illustrate, suppose we invest [latex]\\$2500[\/latex] in an account that offers an annual interest rate of [latex]5\\%[\/latex] compounded continuously. We already know that the balance in our account for any year t can be found with the equation [latex]A=2500e^{-.05t}.[\/latex]<\/p>\n<p id=\"fs-id1165137668181\">But what if we wanted to know the year for any balance? We would need to create a corresponding new function by interchanging the input and the output; thus we would need to create a logarithmic model for this situation. By graphing the model, we can see the output (year) for any input (account balance). For instance, what if we wanted to know how many years it would take for our initial investment to double? Figure 1 shows this point on the logarithmic graph.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_959\" aria-describedby=\"caption-attachment-959\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-959\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-1-300x159.webp\" alt=\"\" width=\"300\" height=\"159\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-1-300x159.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-1-768x407.webp 768w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-1-65x34.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-1-225x119.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-1-350x185.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-1.webp 937w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-959\" class=\"wp-caption-text\">Figure 1<\/figcaption><\/figure>\n<p id=\"fs-id1165135161452\">In this section we will discuss the values for which a logarithmic function is defined, and then turn our attention to graphing the family of logarithmic functions.<\/p>\n<section id=\"fs-id1165137923503\" data-depth=\"1\">\n<h2 data-type=\"title\">Finding the Domain of a Logarithmic Function<\/h2>\n<p id=\"fs-id1165137748716\">Before working with graphs, we will take a look at the domain (the set of input values) for which the logarithmic function is defined.<\/p>\n<p id=\"fs-id1165137758495\">Recall that the exponential function is defined as [latex]y=b^x[\/latex] for any real number <em>x<\/em> and constant [latex]b> 0, b\\not=1,[\/latex] where<\/p>\n<ul id=\"fs-id1165137736024\">\n<li>The domain of <em>y<\/em> is [latex](-\\infty, \\infty).[\/latex]<\/li>\n<li>The range of <em>y<\/em> is [latex](0, \\infty).[\/latex]<\/li>\n<\/ul>\n<p id=\"fs-id1165135641666\">In the last section we learned that the logarithmic function [latex]y=\\log_b(x)[\/latex] is the inverse of the exponential function [latex]y=b^x.[\/latex] So, as inverse functions:<\/p>\n<ul id=\"fs-id1165137656096\">\n<li>The domain of [latex]y=\\log_b(x)[\/latex] is the range of [latex]y=b^x: (0, \\infty).[\/latex]<\/li>\n<li>The range of [latex]y=\\log_b(x)[\/latex] is the domain of [latex]y=b_x: (-\\infty, \\infty).[\/latex]<\/li>\n<\/ul>\n<p id=\"fs-id1165135245571\">Transformations of the parent function [latex]y=\\log_b(x)[\/latex] behave similarly to those of other functions. Just as with other parent functions, we can apply the four types of transformations\u2014shifts, stretches, compressions, and reflections.<\/p>\n<p id=\"fs-id1165137653624\">In Graphs of Exponential Functions we saw that certain transformations can change the <em data-effect=\"italics\">range<\/em> of [latex]y=b^x.[\/latex] Similarly, applying transformations to the parent function [latex]y=\\log_b(x)[\/latex] can change the <em data-effect=\"italics\">domain<\/em>. When finding the domain of a logarithmic function, therefore, it is important to remember that the domain consists <em data-effect=\"italics\">only of positive real numbers<\/em>. That is, the argument of the logarithmic function must be greater than zero.<\/p>\n<p id=\"fs-id1165137851584\">For example, consider [latex]f(x)=\\log_4(2x-3).[\/latex] This function is defined for any values of <em>x<\/em> such that the argument, in this case [latex]2x-3,[\/latex] is greater than zero. To find the domain, we set up an inequality and solve for <em>x<\/em>.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lll} 2x-3 &>& 0 && \\text{Show the argument greater than zero.} \\\\ 2x &>& 3 && \\text{add 3.} \\\\ x &>& 1.5 && \\text{Divide by 2.} \\end{array}[\/latex]<\/p>\n<p id=\"fs-id1165137645047\">In interval notation, the domain of [latex]f(x)\\log_4(2x-3)[\/latex] is [latex](1.5, \\infty).[\/latex]<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Given a logarithmic function, identify the domain.<\/strong><\/p>\n<ol>\n<li>Set up an inequality showing the argument greater than zero.<\/li>\n<li>Solve for <em>x<\/em>.<\/li>\n<li>Write the domain in interval notation.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 1: Identifying the Domain of a Logarithmic Shift<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>What is the domain of [latex]f(x)=\\log_2(x+3)?[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>The logarithmic function is defined only when the input is positive, so this function is defined when [latex]x+3>0.[\/latex] Solving this inequality,<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lll} x+3 &>& 0 && \\text{The input must be positive.} \\\\ x &>& -3 && \\text{Subtract 3.} \\end{array}[\/latex]<\/p>\n<p>The domain of [latex]f(x)=\\log_2(x+3)[\/latex] is [latex](-3, \\infty).[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #1<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>What is the domain of [latex]f(x)=\\log_5(x-2)+1?[\/latex]<\/p>\n<\/div>\n<\/div>\n<section>\n<div class=\"os-note-body\">\n<div id=\"ti_04_04_01\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165134274544\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 2: Identifying the Domain of a Logarithmic Shift and Reflection<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>What is the domain of [latex]f(x)=\\log(5-2x)?[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>The logarithmic function is defined only when the input is positive, so this function is defined when [latex]5-2x>0.[\/latex] Solving this inequality,<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lll} 5-2x &>& 0 && \\text{The input must be positive.} \\\\ -2x &>& -5 && \\text{Subtract 5.} \\\\ x &<& \\frac{5}{2} && \\text{Divide by -2 and switch the inequality.} \\end{array}[\/latex]<\/p>\n<p>The domain of [latex]f(x)=\\log(5-2x)[\/latex] is [latex]\\left(-\\infty, \\frac{5}{2}\\right).[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/section>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>What is the domain of [latex]f(x)=\\log(x-5)+2?[\/latex]<\/p>\n<\/div>\n<\/div>\n<section id=\"fs-id1165137554026\" data-depth=\"1\">\n<h2 data-type=\"title\">Graphing Logarithmic Functions<\/h2>\n<p id=\"fs-id1165134104063\">Now that we have a feel for the set of values for which a logarithmic function is defined, we move on to graphing logarithmic functions. The family of logarithmic functions includes the parent function [latex]y=\\log_b(x)[\/latex] along with all its transformations: shifts, stretches, compressions, and reflections.<\/p>\n<p id=\"fs-id1165137679088\">We begin with the parent function [latex]y=\\log_b(x).[\/latex] Because every logarithmic function of this form is the inverse of an exponential function with the form [latex]y=b^x,[\/latex] their graphs will be reflections of each other across the line [latex]y=x.[\/latex] To illustrate this, we can observe the relationship between the input and output values of [latex]y=2^x[\/latex] and its equivalent [latex]x=\\log_2(y)[\/latex] in Table 1.<\/p>\n<div id=\"Table_04_04_01\" class=\"os-table\">\n<table class=\"grid\" data-id=\"Table_04_04_01\">\n<caption>Table 1<\/caption>\n<tbody>\n<tr>\n<td data-align=\"center\">[latex]x[\/latex]<\/td>\n<td data-align=\"center\">[latex]-3[\/latex]<\/td>\n<td data-align=\"center\">[latex]-2[\/latex]<\/td>\n<td data-align=\"center\">[latex]-1[\/latex]<\/td>\n<td data-align=\"center\">[latex]0[\/latex]<\/td>\n<td data-align=\"center\">[latex]1[\/latex]<\/td>\n<td data-align=\"center\">[latex]2[\/latex]<\/td>\n<td data-align=\"center\">[latex]3[\/latex]<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">[latex]2^x=y[\/latex]<\/td>\n<td data-align=\"center\">[latex]\\frac{1}{8}[\/latex]<\/td>\n<td data-align=\"center\">[latex]\\frac{1}{4}[\/latex]<\/td>\n<td data-align=\"center\">[latex]\\frac{1}{2}[\/latex]<\/td>\n<td data-align=\"center\">[latex]1[\/latex]<\/td>\n<td data-align=\"center\">[latex]2[\/latex]<\/td>\n<td data-align=\"center\">[latex]4[\/latex]<\/td>\n<td data-align=\"center\">[latex]8[\/latex]<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">[latex]\\log_2(y)=x[\/latex]<\/td>\n<td data-align=\"center\">[latex]-3[\/latex]<\/td>\n<td data-align=\"center\">[latex]-2[\/latex]<\/td>\n<td data-align=\"center\">[latex]-1[\/latex]<\/td>\n<td data-align=\"center\">[latex]0[\/latex]<\/td>\n<td data-align=\"center\">[latex]1[\/latex]<\/td>\n<td data-align=\"center\">[latex]2[\/latex]<\/td>\n<td data-align=\"center\">[latex]3[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"os-caption-container\">\n<p>Using the inputs and outputs from Table 1, we can build another table to observe the relationship between points on the graphs of the inverse functions [latex]f(x)=2^x[\/latex] and [latex]g(x)=\\log_2(x).[\/latex] See Table 2.<\/p>\n<div id=\"Table_04_04_02\" class=\"os-table\">\n<table class=\"grid\" data-id=\"Table_04_04_02\">\n<caption>Table 2<\/caption>\n<tbody>\n<tr>\n<td data-align=\"center\">[latex]f(x)=2^x[\/latex]<\/td>\n<td data-align=\"center\">[latex](-3, \\frac{1}{8})[\/latex]<\/td>\n<td data-align=\"center\">[latex](-2, \\frac{1}{4})[\/latex]<\/td>\n<td data-align=\"center\">[latex](-1, \\frac{1}{2})[\/latex]<\/td>\n<td data-align=\"center\">[latex](0, 1)[\/latex]<\/td>\n<td data-align=\"center\">[latex](1, 2)[\/latex]<\/td>\n<td data-align=\"center\">[latex](2, 4)[\/latex]<\/td>\n<td data-align=\"center\">[latex](3, 8)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">[latex]g(x)=\\log_2(x)[\/latex]<\/td>\n<td data-align=\"center\">[latex](\\frac{1}{8}, -3)[\/latex]<\/td>\n<td data-align=\"center\">[latex](\\frac{1}{4}, -2)[\/latex]<\/td>\n<td data-align=\"center\">[latex](\\frac{1}{2}, -1)[\/latex]<\/td>\n<td data-align=\"center\">[latex](1, 0)[\/latex]<\/td>\n<td data-align=\"center\">[latex](2, 1)[\/latex]<\/td>\n<td data-align=\"center\">[latex](4, 2)[\/latex]<\/td>\n<td data-align=\"center\">[latex](8, 3)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"os-caption-container\">\n<p>As we\u2019d expect, the <em data-effect=\"italics\">x<\/em>&#8211; and <em data-effect=\"italics\">y<\/em>-coordinates are reversed for the inverse functions. Figure 2 shows the graph of <em>f<\/em> and <em>g<\/em>.<\/p>\n<figure id=\"attachment_961\" aria-describedby=\"caption-attachment-961\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-961\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-2-300x270.webp\" alt=\"\" width=\"300\" height=\"270\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-2-300x270.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-2-65x58.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-2-225x202.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-2-350x315.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-2.webp 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-961\" class=\"wp-caption-text\">Figure 4. Notice that the graphs of [latex] f(x)=2^x [\/latex] and [latex] g(x)=\\log_2(x) [\/latex] are reflections about the line [latex] y=x. [\/latex]<\/figcaption><\/figure>\n<p id=\"fs-id1165137406913\">Observe the following from the graph:<\/p>\n<ul id=\"fs-id1165137408405\">\n<li>[latex]f(x)=2^x[\/latex] has a <em data-effect=\"italics\">y<\/em>-intercept at [latex](0, 1)[\/latex] and [latex]g(x)=\\log_2(x)[\/latex] has an <em data-effect=\"italics\">x<\/em>&#8211; intercept at [latex](1, 0).[\/latex]<\/li>\n<li>The domain of [latex]f(x)=2^x, (-\\infty, \\infty),[\/latex] is the same as the range of [latex]g(x)=\\log_2(x).[\/latex]<\/li>\n<li>The range of [latex]f(x)=2^x, (0, \\infty),[\/latex] is the same as the domain of [latex]g(x)=\\log_2(x).[\/latex]<\/li>\n<\/ul>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\">Characteristics of the Graph of the Parent Function, [latex]f(x)=\\log_b(x):[\/latex]<\/span><\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>For any real number x and constant [latex]b> 0, b\\not=1,[\/latex]\u00a0we can see the following characteristics in the graph of [latex]f(x)=  log_b(x):[\/latex]<\/p>\n<ul>\n<li>one-to-one function<\/li>\n<li>vertical asymptote: [latex]x=0[\/latex]<\/li>\n<li>domain: [latex](0, \\infty)[\/latex]<\/li>\n<li>range: [latex](-\\infty, \\infty)[\/latex]<\/li>\n<li><em data-effect=\"italics\">x-<\/em>intercept: [latex](1, 0)[\/latex] and key point [latex](b, 1)[\/latex]<\/li>\n<li><em data-effect=\"italics\">y<\/em>-intercept: none<\/li>\n<li>increasing if [latex]b> 1[\/latex]<\/li>\n<li>decreasing if [latex]0< b< 1[\/latex]<\/li>\n<\/ul>\n<p>See Figure 3.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_962\" aria-describedby=\"caption-attachment-962\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-962\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-3-300x134.webp\" alt=\"\" width=\"300\" height=\"134\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-3-300x134.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-3-768x342.webp 768w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-3-65x29.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-3-225x100.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-3-350x156.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-3.webp 824w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-962\" class=\"wp-caption-text\">Figure 3<\/figcaption><\/figure>\n<p>Figure 4 shows how changing the base <em>b<\/em> in [latex]f(x)=\\log_b(x)[\/latex] can affect the graphs. Observe that the graphs compress vertically as the value of the base increases. (<em data-effect=\"italics\">Note:<\/em> recall that the function [latex]\\ln(x)[\/latex] has base [latex]e\\approx 2.718.)[\/latex]<\/p>\n<figure id=\"attachment_963\" aria-describedby=\"caption-attachment-963\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-963\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-4-300x224.webp\" alt=\"\" width=\"300\" height=\"224\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-4-300x224.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-4-65x48.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-4-225x168.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-4-350x261.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-4.webp 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-963\" class=\"wp-caption-text\">Figure 4. The graphs of three logarithmic functions with different bases, all greater than 1.<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Given a logarithmic function with the form [latex]f(x)=\\log_b(x),[\/latex] graph the function <\/strong><\/p>\n<ol>\n<li>Draw and label the vertical asymptote, [latex]x=0.[\/latex]<\/li>\n<li>Plot the <em data-effect=\"italics\">x-<\/em>intercept, [latex](1, 0).[\/latex]<\/li>\n<li>Plot the key point [latex](b, 0).[\/latex]<\/li>\n<li>Draw a smooth curve through the points.<\/li>\n<li>State the domain, [latex](0, \\infty),[\/latex] the range, [latex](-\\infty, \\infty),[\/latex] and the vertical asymptote, [latex]x=0.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 3: Graphing a Logarithmic Function with the Form [latex]f(x)=\\log_b(x).[\/latex].<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Graph [latex]f(x)=\\log_5(x).[\/latex] State the domain, range, and asymptote.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Before graphing, identify the behavior and key points for the graph.<\/p>\n<ul>\n<li>Since\u00a0[latex]b=5[\/latex] is greater than one, we know the function is increasing. The left tail of the graph will approach the vertical asymptote [latex]x=0[\/latex] and the right tail will increase slowly without bound.<\/li>\n<li>The <em data-effect=\"italics\">x<\/em>-intercept is [latex](1, 0).[\/latex]<\/li>\n<li>The key point [latex](5, 1)[\/latex] is on the graph.<\/li>\n<li>We draw and label the asymptote, plot and label the points, and draw a smooth curve through the points (see Figure 5).<\/li>\n<\/ul>\n<figure id=\"attachment_964\" aria-describedby=\"caption-attachment-964\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-964\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-5-300x226.webp\" alt=\"\" width=\"300\" height=\"226\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-5-300x226.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-5-65x49.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-5-225x170.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-5-350x264.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-5.webp 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-964\" class=\"wp-caption-text\">Figure 5<\/figcaption><\/figure>\n<p>The domain is [latex](0, \\infty),[\/latex] the range is [latex](-\\infty, \\infty),[\/latex] and the vertical asymptote is [latex]x=0.[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #3<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Graph [latex]f(x)=\\log_{\\frac{1}{5}}(x).[\/latex] State the domain, range, and asymptote.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165137430980\" data-depth=\"1\">\n<h2 data-type=\"title\">Graphing Transformations of Logarithmic Functions<\/h2>\n<p id=\"fs-id1165137430986\">As we mentioned in the beginning of the section, transformations of logarithmic graphs behave similarly to those of other parent functions. We can shift, stretch, compress, and reflect the parent function [latex]y=\\log_b(x)[\/latex] without loss of shape.<\/p>\n<section id=\"fs-id1165137734884\" data-depth=\"2\">\n<h3 data-type=\"title\">Graphing a Horizontal Shift of <em data-effect=\"italics\">f<\/em>(<em data-effect=\"italics\">x<\/em>) = log<sub><em data-effect=\"italics\">b<\/em><\/sub>(<em data-effect=\"italics\">x<\/em>)<\/h3>\n<p id=\"fs-id1165135530294\">When a constant <em>c<\/em> is added to the input of the parent function [latex]f(x)=\\log_b(x),[\/latex] the result is a horizontal shift <em>c<\/em> units in the <em data-effect=\"italics\">opposite<\/em> direction of the sign on <em>c<\/em>. To visualize horizontal shifts, we can observe the general graph of the parent function [latex]f(x)=\\log_b(x)[\/latex] and for [latex]c> 0[\/latex] alongside the shift left, [latex]g(x)=\\log_b(x+c),[\/latex] and the shift right, [latex]h(x)=\\log_b(x-c).[\/latex] See Figure 6.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_965\" aria-describedby=\"caption-attachment-965\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-965\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-6-300x175.webp\" alt=\"\" width=\"300\" height=\"175\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-6-300x175.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-6-768x449.webp 768w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-6-65x38.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-6-225x132.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-6-350x205.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-6.webp 975w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-965\" class=\"wp-caption-text\">Figure 6<\/figcaption><\/figure>\n<div id=\"fs-id1165135296307\" class=\"ui-has-child-title\" data-type=\"note\">\n<header>\n<div class=\"textbox textbox--examples\"><\/div>\n<\/header>\n<header class=\"textbox__header\">\n<p class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\">Horizontal Shifts of the Parent Function [latex]f(x)=\\log_b(c)[\/latex]<\/span><\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>For any constant <em>c<\/em>, the function [latex]f(x)=\\log_b(x+c)[\/latex]<\/p>\n<ul>\n<li>shifts the parent function [latex]y=\\log_b(x)[\/latex] left c units if [latex]c> 0.[\/latex]<\/li>\n<li>shifts the parent function [latex]y=\\log_b(x)[\/latex] right c units if [latex]c< 0.[\/latex]<\/li>\n<li>has the vertical asymptote [latex]x=-c.[\/latex]<\/li>\n<li>has domain [latex](-c, \\infty).[\/latex]<\/li>\n<li>has range [latex](-\\infty, \\infty).[\/latex]<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Given a logarithmic function with the form [latex]f(x)=\\log_b(x+c),[\/latex] graph the translation. <\/strong><\/p>\n<ol>\n<li>Identify the horizontal shift:\n<ol>\n<li>If [latex]c> 0,[\/latex] shift the graph of [latex]f(x)=\\log_b(x)[\/latex] left <em>c<\/em> units.<\/li>\n<li>If [latex]c< 0,[\/latex] shift the graph of [latex]f(x)=\\log_b(x)[\/latex] right <em>c<\/em> units.<\/li>\n<\/ol>\n<\/li>\n<li>Draw the vertical asymptote [latex]x=-c.[\/latex]<\/li>\n<li>Identify three key points from the parent function. Find new coordinates for the shifted functions by subtracting c from the x-coordinate.<\/li>\n<li>Label the three points.<\/li>\n<li>The Domain is [latex](-c, \\infty),[\/latex] the range is [latex](-\\infty, \\infty),[\/latex] and the vertical asymptote is [latex]x=-c.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 4:Graphing a Horizontal Shift of the Parent Function [latex]y=\\log_b(x)[\/latex]<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Sketch the horizontal shift [latex]f(x)=\\log_3(x-2)[\/latex] alongside its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Since the function is [latex]f(x)=\\log_3(x-2),[\/latex] we notice x+(-2)=x-2.<\/p>\n<p>Thus\u00a0 [latex]c=-2,[\/latex] so [latex]c< 0.[\/latex] This means we will shift the function [latex]f(x)=\\log_3(x)[\/latex] right 2 units.\n\nThe vertical asymptote is [latex]x=-(-2)[\/latex] or [latex]x=2.[\/latex]\n\nConsider the three key points from the parent function, [latex]\\left(\\frac{1}{3}, -1\\right), (1, 0),[\/latex] and [latex](3, 1).[\/latex]\n\nThe new coordinates are found by adding 2 to the <em>x<\/em>-coordinates.<\/p>\n<p>Label the points [latex]\\left(\\frac{7}{3}, -1\\right), (3, 0),[\/latex] and [latex](5, 1).[\/latex]<\/p>\n<p>The domain is [latex](2, \\infty),[\/latex] the range is [latex](-\\infty, \\infty),[\/latex] and the vertical asymptote is [latex]x=2.[\/latex]<\/p>\n<figure id=\"attachment_966\" aria-describedby=\"caption-attachment-966\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-966\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-7-300x224.webp\" alt=\"\" width=\"300\" height=\"224\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-7-300x224.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-7-65x48.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-7-225x168.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-7-350x261.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-7.webp 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-966\" class=\"wp-caption-text\">Figure 7<\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #4<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Sketch a graph of [latex]f(x)=\\log_3(x+4)[\/latex] alongside its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165135403538\" data-depth=\"2\">\n<h3 data-type=\"title\">Graphing a Vertical Shift of <em data-effect=\"italics\">y<\/em> = log<sub><em data-effect=\"italics\">b<\/em><\/sub>(<em data-effect=\"italics\">x<\/em>)<\/h3>\n<p id=\"fs-id1165134310784\">When a constant <em>d<\/em> is added to the parent function [latex]f(x)=\\log_b(x),[\/latex] the result is a vertical shift <em>d<\/em> units in the direction of the sign on <em>d<\/em>. To visualize vertical shifts, we can observe the general graph of the parent function [latex]f(x)=\\log_b(x)[\/latex] alongside the shift up, [latex]g(x)=\\log_b(x)+d[\/latex] and the shift down, [latex]h(x)=\\log_b(x)-d.[\/latex] See Figure 8.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_967\" aria-describedby=\"caption-attachment-967\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-967\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-8-300x228.webp\" alt=\"\" width=\"300\" height=\"228\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-8-300x228.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-8-768x584.webp 768w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-8-65x49.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-8-225x171.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-8-350x266.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-8.webp 975w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-967\" class=\"wp-caption-text\">Figure 8<\/figcaption><\/figure>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\">Vertical Shifts of the Parent Function [latex]y=\\log_b(x)[\/latex]<\/span><\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>For any constant d, the function [latex]f(x)=\\log_b(x)+d[\/latex]<\/p>\n<ul>\n<li>shifts the parent function [latex]f(x)=\\log_b(x)[\/latex] up d units if [latex]d> 0.[\/latex]<\/li>\n<li>shifts the parent function [latex]f(x)=\\log_b(x)[\/latex] down d units if [latex]d< 0.[\/latex]<\/li>\n<li>has the vertical asymptote [latex]x=0.[\/latex]<\/li>\n<li>has domain [latex](0, \\infty).[\/latex]<\/li>\n<li>has range [latex](-\\infty, \\infty).[\/latex]<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Given a logarithmic function with the form [latex]f(x)=\\log_b(x)+d,[\/latex] graph the translation. <\/strong><\/p>\n<ol>\n<li>Identify the vertical shift:\n<ol>\n<li>If [latex]d> 0,[\/latex] shift the graph of [latex]f(x)=\\log_b(x)[\/latex] up d units.<\/li>\n<li>If [latex]d< 0,[\/latex] shift the graph of [latex]f(x)=\\log_b(x)[\/latex] down d units.<\/li>\n<\/ol>\n<\/li>\n<li>Draw the vertical asymptote [latex]x=0.[\/latex]<\/li>\n<li>Identify three key points from the parent function. Find new coordinates for the shifted functions by adding d to the y-coordinate.<\/li>\n<li>Label the three points.<\/li>\n<li>The domain is [latex](0, \\infty),[\/latex] the range is [latex](-\\infty, \\infty),[\/latex] and the vertical asymptote is [latex]x=0.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 5: Graphing a Vertical Shift of the Parent Function [latex]y=\\log_b(x)[\/latex]<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Sketch a graph of [latex]f(x)=\\log_3(x)-2[\/latex] alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Since the function is [latex]f(x)=\\log_3(x)-2,[\/latex] we will notice [latex]d=-2.[\/latex] Thus [latex]d< 0.[\/latex]\n\nThis means we will shift the function [latex]f(x)=\\log_3(2)[\/latex] down 2 units.\n\nThe vertical asymptote is [latex]x=0.[\/latex]\n\nConsider the three key points from the parent function, [latex]\\left(\\frac{1}{3}, -1\\right), (1, 0),[\/latex] and [latex](3, 1).[\/latex]\n\n\n<p id=\"fs-id1165135503945\">The new coordinates are found by subtracting 2 from the <em data-effect=\"italics\">y <\/em>coordinates.<\/p>\n<p>Label the points [latex]\\left(\\frac{1}{3}, -3\\right), (1, -2),[\/latex]\u00a0and [latex](3, -1).[\/latex]<\/p>\n<figure id=\"attachment_968\" aria-describedby=\"caption-attachment-968\" style=\"width: 283px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-968\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-9-283x300.webp\" alt=\"\" width=\"283\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-9-283x300.webp 283w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-9-65x69.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-9-225x238.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-9-350x371.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-9.webp 487w\" sizes=\"auto, (max-width: 283px) 100vw, 283px\" \/><figcaption id=\"caption-attachment-968\" class=\"wp-caption-text\">Figure 9<\/figcaption><\/figure>\n<p>The domain is [latex](0, \\infty),[\/latex] the range is [latex](-\\infty, \\infty),[\/latex] and the vertical asymptote is [latex]x=0.[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #5<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Sketch a graph of [latex]f(x)=\\log_2(x)+2[\/latex] alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165137770245\" data-depth=\"2\">\n<h3 data-type=\"title\">Graphing Stretches and Compressions of <em data-effect=\"italics\">y<\/em> = log<sub><em data-effect=\"italics\">b<\/em><\/sub>(<em data-effect=\"italics\">x<\/em>)<\/h3>\n<p id=\"fs-id1165137418005\">When the parent function [latex]f(x)=\\log_b(x)[\/latex] is multiplied by a constant [latex]a> 0[\/latex] the result is a vertical stretch or compression of the original graph. To visualize stretches and compressions, we set [latex]a> 1[\/latex] and observe the general graph of the parent function [latex]f(x)=\\log_b(x)[\/latex] alongside the vertical stretch, [latex]g(x)=a\\log_b(x)[\/latex] and the vertical compression, [latex]h(x)=\\frac{1}{a}\\log_b(x).[\/latex] See Figure 10.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_969\" aria-describedby=\"caption-attachment-969\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-969\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-10-300x233.webp\" alt=\"\" width=\"300\" height=\"233\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-10-300x233.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-10-768x597.webp 768w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-10-65x51.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-10-225x175.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-10-350x272.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-10.webp 975w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-969\" class=\"wp-caption-text\">Figure 10<\/figcaption><\/figure>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\">Vertical Stretches and Compressions of the Parent Function- [latex]y=\\log_b(x)[\/latex]<\/span><\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>For any constant [latex]a> 1,[\/latex] the function [latex]f(x)=a\\log_b(x)[\/latex]<\/p>\n<ul>\n<li>stretches the parent function [latex]y=\\log_b(x)[\/latex] vertically by a factor of a if [latex]a> 1.[\/latex]<\/li>\n<li>compresses the parent function [latex]y=\\log_b(x)[\/latex] vertically by a factor of a if [latex]0< a< 1.[\/latex]<\/li>\n<li>has the vertical asymptote [latex]x=0.[\/latex]<\/li>\n<li>has the <em data-effect=\"italics\">x<\/em>-intercept [latex](1, 0).[\/latex]<\/li>\n<li>has domain [latex](0, \\infty).[\/latex]<\/li>\n<li>has range [latex](-\\infty, \\infty).[\/latex]<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Given a logarithmic function with the form [latex]f(x)=a\\log_b(x), a> 0,[\/latex] graph the translation. <\/strong><\/p>\n<ol>\n<li>Identify the vertical stretch or compressions:\n<ol>\n<li>If [latex]|a|> 1,[\/latex] the graph of [latex]f(x)=\\log_b(x)[\/latex] is stretched by a factor of a units.<\/li>\n<li>If [latex]|a|< 1,[\/latex] the graph of [latex]f(x)=\\log_b(x)[\/latex] is compressed by a factor of a units.<\/li>\n<\/ol>\n<\/li>\n<li>Draw the vertical asymptote [latex]x=0.[\/latex]<\/li>\n<li>Identify three key points from the parent function. Find new coordinates for the shifted functions by multiplying the <em>y<\/em>-coordinates by <em>a<\/em>.<\/li>\n<li>Label the three points.<\/li>\n<li>The domain is [latex](0, \\infty),[\/latex] the range is [latex](-\\infty, \\infty),[\/latex] and the vertical asymptote is [latex]x=0.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 6: Graphing a Stretch or Compression of the Parent Function [latex]y=\\log_b(x)[\/latex]<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Sketch a graph of [latex]f(x)=2\\log_4(x)[\/latex] alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Since the function is [latex]f(x)=2\\log_4(x),[\/latex] we will notice [latex]a=2.[\/latex]<\/p>\n<p>This means we will stretch the function [latex]f(x)=\\log_4(x)[\/latex] by a factor of 2.<\/p>\n<p>The vertical asymptote is [latex]x=0.[\/latex]<\/p>\n<p>Consider the three key points from the parent function, [latex]\\left(\\frac{1}{4}, -1\\right), (1, 0),[\/latex] and [latex](4, 1).[\/latex]<\/p>\n<p id=\"fs-id1165135570058\">The new coordinates are found by multiplying the <em>y<\/em>-coordinates by 2.<\/p>\n<p>Label the points [latex]\\left(\\frac{1}{4}, -2\\right), (1, 0),[\/latex] and [latex](4, 2).[\/latex]<\/p>\n<p>The domain is [latex](0, \\infty),[\/latex] the range is [latex](-\\infty, \\infty),[\/latex] and the vertical asymptote is [latex]x=0.[\/latex] See Figure 11.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_970\" aria-describedby=\"caption-attachment-970\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-970\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-11-300x225.webp\" alt=\"\" width=\"300\" height=\"225\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-11-300x225.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-11-65x49.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-11-225x169.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-11-350x263.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-11.webp 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-970\" class=\"wp-caption-text\">Figure 11<\/figcaption><\/figure>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #6<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Sketch a graph of [latex]f(x)=\\frac{1}{2}\\log_4(x)[\/latex] alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 7: Combining a Shift and a Stretch<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Sketch a graph of [latex]f(x)=5\\log(x+2).[\/latex] State the domain, range, and asymptote.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Remember: what happens inside parentheses happens first. First, we move the graph left 2 units, then stretch the function vertically by a factor of 5, as in Figure 12. The vertical asymptote will be shifted to [latex]x=-2.[\/latex] The <em data-effect=\"italics\">x<\/em>-intercept will be [latex](-1, 0).[\/latex] The domain will be [latex](-2, \\infty).[\/latex] Two points will help give the shape of the graph: [latex](-1, 0)[\/latex] and [latex](8, 5).[\/latex] We chose [latex]x=8[\/latex] as the <em data-effect=\"italics\">x<\/em>-coordinate of one point to graph because when [latex]x=8, x+2=10,[\/latex]\u00a0the base of the common logarithm.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_971\" aria-describedby=\"caption-attachment-971\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-971\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-12-300x272.webp\" alt=\"\" width=\"300\" height=\"272\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-12-300x272.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-12-65x59.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-12-225x204.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-12-350x317.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-12.webp 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-971\" class=\"wp-caption-text\">Figure 12<\/figcaption><\/figure>\n<p>The domain is [latex](-2, \\infty)[\/latex] the range is [latex](-\\infty, \\infty),[\/latex] and the vertical asymptote is [latex]x=-2.[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #7<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Sketch a graph of the function [latex]f(x)=3\\log(x-2)+1.[\/latex] State the domain, range, and asymptote.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165137629003\" data-depth=\"2\">\n<h3 data-type=\"title\">Graphing Reflections of [latex]f(x)=\\log_b(x)[\/latex]<\/h3>\n<p id=\"fs-id1165135169315\">When the parent function [latex]f(x)=\\log_b(x)[\/latex] is multiplied by -1, the result is a reflection about the <em data-effect=\"italics\">x<\/em>-axis. When the <em data-effect=\"italics\">input<\/em> is multiplied by -1, the result is a reflection about the <em data-effect=\"italics\">y<\/em>-axis. To visualize reflections, we restrict [latex]b> 1,[\/latex] and observe the general graph of the parent function [latex]f(x)=\\log_b(x)[\/latex] alongside the reflection about the <em data-effect=\"italics\">x<\/em>-axis, [latex]g(x)=-\\log_b(x)[\/latex] and the reflection about the <em data-effect=\"italics\">y<\/em>-axis, [latex]h(x)=log_b(-x).[\/latex]<\/p>\n<figure id=\"attachment_972\" aria-describedby=\"caption-attachment-972\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-972\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-13-300x242.webp\" alt=\"\" width=\"300\" height=\"242\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-13-300x242.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-13-768x619.webp 768w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-13-65x52.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-13-225x181.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-13-350x282.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-13.webp 975w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-972\" class=\"wp-caption-text\">Figure 13<\/figcaption><\/figure>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"os-title\" data-type=\"title\"><span class=\"os-title-label\" data-type=\"\">Reflections of the Parent Function [latex]y=\\log_b(x)[\/latex]<\/span><\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>The function [latex]f(x)=-\\log_b(x)[\/latex]<\/p>\n<ul>\n<li>reflects the parent function [latex]y=\\log_b(x)[\/latex] about the <em data-effect=\"italics\">x<\/em>-axis.<\/li>\n<li>has domain, [latex](0, \\infty),[\/latex] range, [latex](-\\infty, \\infty),[\/latex] and vertical asymptote, [latex]x=0,[\/latex] which are unchanged from the parent function.<\/li>\n<\/ul>\n<p>The function [latex]f(x)=\\log_b(-x)[\/latex]<\/p>\n<ul>\n<li>reflects the parent function [latex]y=\\log_b(x)[\/latex] about the <em data-effect=\"italics\">y<\/em>-axis.<\/li>\n<li>has domain [latex](-\\infty, 0).[\/latex]<\/li>\n<li>has range, [latex](-\\infty, \\infty),[\/latex] and vertical asymptote, [latex]x=0,[\/latex] which are unchanged from the parent function.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Given a logarithmic function with the parent function [latex]f(x)=\\log_b(x),[\/latex] graph a translation. <\/strong><\/p>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%; height: 160px;\">\n<caption>Table 3<\/caption>\n<tbody>\n<tr style=\"height: 20px;\">\n<td style=\"width: 50%; height: 20px;\">If [latex]f(x)=-\\log_b(x)[\/latex]<\/td>\n<td style=\"width: 50%; height: 20px;\">If [latex]f(x)=\\log_b(-x)[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 47px;\">\n<td style=\"width: 50%; height: 47px;\">1. Draw the vertical asymptote, [latex]x=0.[\/latex]<\/td>\n<td style=\"width: 50%; height: 47px;\">1. Draw the vertical asymptote, [latex]x=0.[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 48px;\">\n<td style=\"width: 50%; height: 48px;\">2. Plot the <em data-effect=\"italics\">x-<\/em>intercept, [latex](1, 0).[\/latex]<\/td>\n<td style=\"width: 50%; height: 48px;\">2. Plot the <em data-effect=\"italics\">x-<\/em>intercept, [latex](1, 0).[\/latex]<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 50%; height: 15px;\">3. Reflect the graph of the parent function [latex]f(x)=\\log_b(x)[\/latex] about the <em data-effect=\"italics\">x<\/em>-axis.<\/td>\n<td style=\"width: 50%; height: 15px;\">3. Reflect the graph of the parent function [latex]f(x)=\\log_b(x)[\/latex] about the <em data-effect=\"italics\">y<\/em>-axis.<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 50%;\" data-align=\"left\">4. Draw a smooth curve through the points.<\/td>\n<td style=\"width: 50%; height: 15px;\">4. Draw a smooth curve through the points.<\/td>\n<\/tr>\n<tr style=\"height: 15px;\">\n<td style=\"width: 50%; height: 15px;\">5. State the domain, [latex](0, \\infty),[\/latex] the range, [latex](-\\infty, \\infty),[\/latex] and the vertical asymptote [latex]x=0.[\/latex]<\/td>\n<td style=\"width: 50%; height: 15px;\">5. State the domain, [latex](-\\infty, 0),[\/latex] the range, [latex](-\\infty, \\infty),[\/latex] and the vertical asymptote [latex]x=0.[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 8: Graphing a Reflection of a Logarithmic Function<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Sketch a graph of [latex]f(x)=\\log(-x)[\/latex] alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Before graphing [latex]f(x)=\\log(-x),[\/latex] identify the behavior and key points for the graph.<\/p>\n<ul>\n<li>Since [latex]b=10[\/latex] is greater than one, we know that the parent function is increasing. Since the <em data-effect=\"italics\">input<\/em> value is multiplied by -1, <em>f<\/em> is a reflection of the parent graph about the <em data-effect=\"italics\">y-<\/em>axis. Thus, [latex]f(x)=\\log(-x)[\/latex] will be decreasing as <em>x<\/em> moves from negative infinity to zero, and the right tail of the graph will approach the vertical asymptote [latex]x=0.[\/latex]<\/li>\n<li>The <em data-effect=\"italics\">x<\/em>-intercept is [latex](-1, 0).[\/latex]<\/li>\n<li>We draw and label the asymptote, plot and label the points, and draw a smooth curve through the points.<\/li>\n<\/ul>\n<figure id=\"attachment_973\" aria-describedby=\"caption-attachment-973\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-973\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-14-300x223.webp\" alt=\"\" width=\"300\" height=\"223\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-14-300x223.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-14-65x48.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-14-225x167.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-14-350x261.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-14.webp 489w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-973\" class=\"wp-caption-text\">Figure 14<\/figcaption><\/figure>\n<p>The domain is [latex](-\\infty, 0),[\/latex] the range is [latex](-\\infty, \\infty),[\/latex] and the vertical asymptote is [latex]x=0.[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #8<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Graph [latex]f(x)=-\\log(-x).[\/latex] State the domain, range, and asymptote.<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">How To<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Given a logarithmic equation, use a graphing calculator to approximate solutions.<\/strong><\/p>\n<ol>\n<li>Press <strong>[Y=]<\/strong>. Enter the given logarithm equation or equations as <strong>Y<sub>1<\/sub>=<\/strong> and, if needed, <strong>Y<sub>2<\/sub>=<\/strong>.<\/li>\n<li>Press <strong>[GRAPH]<\/strong> to observe the graphs of the curves and use <strong>[WINDOW]<\/strong> to find an appropriate view of the graphs, including their point(s) of intersection.<\/li>\n<li>To find the value of <em>x<\/em>, we compute the point of intersection. Press <strong>[2ND] <\/strong>then <strong>[CALC]<\/strong>. Select \u201cintersect\u201d and press <strong>[ENTER]<\/strong> three times. The point of intersection gives the value of <em>x<\/em> for the point(s) of intersection.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 9: Approximating the Solution of a Logarithmic Equation<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Solve [latex]4\\ln(x)+1=-2\\ln(x-1)[\/latex] graphically. Round to the nearest thousandth.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>Press <strong>[Y=]<\/strong> and enter [latex]4\\ln(x)+1[\/latex] next to <strong>Y<sub>1<\/sub><\/strong>=. Then enter [latex]-2\\ln(x-1)[\/latex] next to <strong>Y<sub>2<\/sub>=<\/strong>. For a window, use the values 0 to 5 for <em>x<\/em> and \u201310 to 10 for <em>y<\/em>. Press <strong>[GRAPH]<\/strong>. The graphs should intersect somewhere a little to right of [latex]x=1.[\/latex]<\/p>\n<p>For a better approximation, press <strong>[2ND] <\/strong>then <strong>[CALC]<\/strong>. Select <strong>[5: intersect]<\/strong> and press <strong>[ENTER]<\/strong> three times. The <em data-effect=\"italics\">x<\/em>-coordinate of the point of intersection is displayed as 1.3385297. (Your answer may be different if you use a different window or use a different value for <strong>Guess?<\/strong>) So, to the nearest thousandth, [latex]x\\approx 1.339.[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #9<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Solve [latex]5\\log(x+2)=4-\\log(x)[\/latex] graphically. Round to the nearest thousandth.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165135528930\" data-depth=\"2\">\n<h3 data-type=\"title\">Summarizing Translations of the Logarithmic Function<\/h3>\n<p id=\"fs-id1165135528935\">Now that we have worked with each type of translation for the logarithmic function, we can summarize each in Table 4 to arrive at the general equation for translating exponential functions.<\/p>\n<div id=\"Table_04_04_009\" class=\"os-table\">\n<table class=\"grid\" data-id=\"Table_04_04_009\">\n<caption>Table 4<\/caption>\n<thead>\n<tr>\n<th colspan=\"2\" scope=\"colgroup\">Transformations of the Parent Function [latex]y=\\log_b(x)[\/latex]<\/th>\n<\/tr>\n<tr>\n<th scope=\"col\" data-align=\"center\">Transformation<\/th>\n<th scope=\"col\" data-align=\"center\">Form<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td data-align=\"center\">Shift<\/p>\n<ul id=\"fs-id1165137416971\">\n<li>Horizontally <em>c<\/em> units to the left<\/li>\n<li>Vertically <em>d<\/em> units up<\/li>\n<\/ul>\n<\/td>\n<td data-align=\"center\">[latex]y=\\log_b(x+c)+d[\/latex]<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">Stretch and Compress<\/p>\n<ul id=\"fs-id1165137427553\">\n<li>Stretch if [latex]|a|> 1[\/latex]<\/li>\n<li>Compression if [latex]|a|< 1[\/latex]<\/li>\n<\/ul>\n<\/td>\n<td data-align=\"center\">[latex]y=a\\log_b(x)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">Reflect about the <em data-effect=\"italics\">x<\/em>-axis<\/td>\n<td data-align=\"center\">[latex]y=-\\log_b(x)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">Reflect about the <em data-effect=\"italics\">y<\/em>-axis<\/td>\n<td data-align=\"center\">[latex]y-\\log_b(-x)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td data-align=\"center\">General equation for all translations<\/td>\n<td data-align=\"center\">[latex]y=a\\log_b(x+c)+d[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div class=\"os-caption-container\">\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Translations of Logarithmic Functions<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>All translations of the parent logarithmic function, [latex]y=\\log_b(x),[\/latex] have the form<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=a\\log_b(x+c)+d[\/latex]<\/p>\n<p>where the parent function, [latex]y=\\log_b(x), b> 1,[\/latex] is<\/p>\n<ul>\n<li>shifted vertically up <em>d<\/em> units.<\/li>\n<li>shifted horizontally to the left <em>c<\/em> units.<\/li>\n<li>stretched vertically by a factor of [latex]|a|[\/latex] if [latex]|a|> 0.[\/latex]<\/li>\n<li>compressed vertically by a factor of [latex]|a|[\/latex] if [latex]0< |a|< 1.[\/latex]<\/li>\n<li>reflected about the <em data-effect=\"italics\">x-<\/em>axis when [latex]a< 0.[\/latex]<\/li>\n<\/ul>\n<p>For [latex]f(x)=\\log(-x),[\/latex] the graph of the parent function is reflected about the <em data-effect=\"italics\">y<\/em>-axis.<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 10: Finding the Vertical Asymptote of a Logarithm Graph<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>What is the vertical asymptote of [latex]f(x)=-2\\log_3(x+4)+5?[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>The vertical asymptote is at [latex]x=-4.[\/latex]<\/p>\n<h3>Analysis<\/h3>\n<p>The coefficient, the base, and the upward translation do not affect the asymptote. The shift of the curve 4 units to the left shifts the vertical asymptote to [latex]x=-4.[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #10<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>What is the vertical asymptote of [latex]f(x)=3+\\ln(x-1)?[\/latex]<\/p>\n<\/div>\n<\/div>\n<section>\n<div class=\"os-note-body\">\n<div id=\"ti_04_04_10\" class=\"unnumbered os-hasSolution\" data-type=\"exercise\">\n<section>\n<div id=\"fs-id1165137748692\" data-type=\"problem\">\n<div class=\"os-problem-container\">\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 11: Finding the Equation from a Graph<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Find a possible equation for the common logarithmic function graphed in Figure 15.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_974\" aria-describedby=\"caption-attachment-974\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-974\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-15-300x226.webp\" alt=\"\" width=\"300\" height=\"226\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-15-300x226.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-15-65x49.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-15-225x170.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-15-350x264.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-15.webp 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-974\" class=\"wp-caption-text\">Figure 15<\/figcaption><\/figure>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>This graph has a vertical asymptote at [latex]x=-2[\/latex] and has been vertically reflected. We do not know yet the vertical shift or the vertical stretch. We know so far that the equation will have form:<\/p>\n<p style=\"text-align: center;\">[latex]f(x)=-a\\log(x_2)+k[\/latex]<\/p>\n<p>It appears the graph passes through the points [latex](-1, 1)[\/latex] and [latex](2, -1).[\/latex] Substituting [latex](-1, 1),[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lll} 1&=& -a\\log(-1+2)+k && \\text{Substitute } (-1, 1). \\\\ 1 &=& -a\\log(1)+k && \\text{Arithmetic.} \\\\ 1 &=& k && \\log(1)=0. \\end{array}[\/latex]<\/p>\n<p>Next, substituting in [latex](2, -1),[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lll} -1 &=& -a\\log(2+2)+1 && \\text{Plug in } (2, -1). \\\\ -2 &=& -a\\log(4) && \\text{Arithmetic.} \\\\ a &=& \\frac{2}{\\log(4)} && \\text{Solve for } a. \\end{array}[\/latex]<\/p>\n<p>This gives us the equation [latex]f(x)=-\\frac{2}{\\log(4)}\\log(x+2)+1.[\/latex]<\/p>\n<h3>Analysis<\/h3>\n<p>We can verify this answer by comparing the function values in Table 5 with the points on the graph in Figure 15.<\/p>\n<table class=\"grid\" style=\"border-collapse: collapse; width: 100%;\">\n<caption>Table 5<\/caption>\n<tbody>\n<tr>\n<td style=\"width: 16.6667%;\">[latex]x[\/latex]<\/td>\n<td style=\"width: 16.6667%;\">-1<\/td>\n<td style=\"width: 16.6667%;\">0<\/td>\n<td style=\"width: 16.6667%;\">1<\/td>\n<td style=\"width: 16.6667%;\">2<\/td>\n<td style=\"width: 16.6667%;\">3<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 16.6667%;\">[latex]f(x)[\/latex]<\/td>\n<td style=\"width: 16.6667%;\">1<\/td>\n<td style=\"width: 16.6667%;\">0<\/td>\n<td style=\"width: 16.6667%;\">-0\/58496<\/td>\n<td style=\"width: 16.6667%;\">-1<\/td>\n<td style=\"width: 16.6667%;\">-1.3219<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 16.6667%;\">[latex]x[\/latex]<\/td>\n<td style=\"width: 16.6667%;\">4<\/td>\n<td style=\"width: 16.6667%;\">5<\/td>\n<td style=\"width: 16.6667%;\">6<\/td>\n<td style=\"width: 16.6667%;\">7<\/td>\n<td style=\"width: 16.6667%;\">8<\/td>\n<\/tr>\n<tr>\n<td style=\"width: 16.6667%;\">[latex]f(x)[\/latex]<\/td>\n<td style=\"width: 16.6667%;\">-1.5850<\/td>\n<td style=\"width: 16.6667%;\">-1.8074<\/td>\n<td style=\"width: 16.6667%;\">-2<\/td>\n<td style=\"width: 16.6667%;\">-2.1699<\/td>\n<td style=\"width: 16.6667%;\">-2.3219<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/details>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Try It #11<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Give the equation of the natural logarithm graphed in Figure 16.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_975\" aria-describedby=\"caption-attachment-975\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-975\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-16-300x272.webp\" alt=\"\" width=\"300\" height=\"272\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-16-300x272.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-16-65x59.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-16-225x204.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-16-350x318.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-16.webp 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-975\" class=\"wp-caption-text\">Figure 16<\/figcaption><\/figure>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Q&amp;A<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Q: Is it possible to tell the domain and range and describe the end behavior of a function just by looking at the graph?<\/strong><\/p>\n<p><em data-effect=\"italics\">A: Yes, if we know the function is a general logarithmic function. For example, look at the graph in Figure 16. The graph approaches <\/em> [latex]x=-3[\/latex] (or thereabouts) more and more closely, so [latex]x=-3[\/latex] is, or is very close to, the vertical asymptote. It approaches from the right, so the domain is all points to the right, [latex]\\{x \\mid x > -3\\}.[\/latex] The range, as with all general logarithmic functions, is all real numbers. And we can see the end behavior because the graph goes down as it goes left and up as it goes right. The end behavior is that as [latex]x\\rightarrow -3^+, f(x)\\rightarrow -\\infty[\/latex] and as [latex]x\\rightarrow \\infty, f(x)\\rightarrow \\infty.[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/section>\n<div class=\"textbox textbox--key-takeaways\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Media<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Access these online resources for additional instruction and practice with graphing logarithms.<\/p>\n<ul>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=w1A2ZYmfGco\">Graph an Exponential Function and Logarithmic Function<\/a><\/li>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=GnfclmCE9rE\">Match Graphs with Exponential and Logarithmic Functions<\/a><\/li>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=_Om0ZMzIgUk\">Find the Domain of Logarithmic Functions<\/a><\/li>\n<\/ul>\n<\/div>\n<\/div>\n<\/section>\n<div class=\"os-eos os-section-exercises-container\" data-uuid-key=\".section-exercises\">\n<h2 data-type=\"document-title\" data-rex-keep=\"true\"><span class=\"os-text\">6.4 Section Exercises<\/span><\/h2>\n<section id=\"fs-id1165135189917\" class=\"section-exercises\" data-depth=\"1\">\n<section id=\"fs-id1165135189921\" data-depth=\"2\">\n<h3 data-type=\"title\">Verbal<\/h3>\n<div id=\"fs-id1165135582182\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135582184\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135582182-solution\">1<\/a><span class=\"os-divider\">. <\/span>The inverse of every logarithmic function is an exponential function and vice-versa. What does this tell us about the relationship between the coordinates of the points on the graphs of each?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137855282\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137855284\" data-type=\"problem\">\n<p><span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span>What type(s) of translation(s), if any, affect the range of a logarithmic function?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137855292\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137855294\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137855292-solution\">3<\/a><span class=\"os-divider\">. <\/span>What type(s) of translation(s), if any, affect the domain of a logarithmic function?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137424692\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137424694\" data-type=\"problem\">\n<p><span class=\"os-number\">4<\/span><span class=\"os-divider\">. <\/span>Consider the general logarithmic function [latex]f(x)=\\log_b(x).[\/latex] Why can\u2019t <em>x<\/em> be zero?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137697126\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137697128\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137697126-solution\">5<\/a><span class=\"os-divider\">. <\/span>Does the graph of a general logarithmic function have a horizontal asymptote? Explain.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<section id=\"fs-id1165137459922\" data-depth=\"2\">\n<h3 data-type=\"title\">Algebraic<\/h3>\n<p id=\"fs-id1165137459927\">For the following exercises, state the domain and range of the function.<\/p>\n<div id=\"fs-id1165137459930\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137737591\" data-type=\"problem\">\n<p><span class=\"os-number\">6<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=\\log_3(x+4)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135264771\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135264773\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135264771-solution\">7<\/a><span class=\"os-divider\">. <\/span> [latex]h(x)=\\ln\\left(\\frac{1}{2}-x\\right)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135253806\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135253808\" data-type=\"problem\">\n<p><span class=\"os-number\">8<\/span><span class=\"os-divider\">. <\/span> [latex]g(x)=\\log_5(2x+9)-2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135415813\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135415815\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135415813-solution\">9<\/a><span class=\"os-divider\">. <\/span> [latex]h(x)=\\ln(4x+17)-5[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137843712\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137843714\" data-type=\"problem\">\n<p><span class=\"os-number\">10<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=\\log_2(12-3x)-3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165135503696\">For the following exercises, state the domain and the vertical asymptote of the function.<\/p>\n<div id=\"fs-id1165134037565\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134037567\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165134037565-solution\">11<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=\\log_b(x-5)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137933908\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137933910\" data-type=\"problem\">\n<p><span class=\"os-number\">12<\/span><span class=\"os-divider\">. <\/span> [latex]g(x)=\\ln(3-x)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137737027\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137737029\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137737027-solution\">13<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=\\log(3x+1)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135188081\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135188083\" data-type=\"problem\">\n<p><span class=\"os-number\">14<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=3\\log(-x)+2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134188310\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134188313\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165134188310-solution\">15<\/a><span class=\"os-divider\">. <\/span> [latex]g(x)=-\\ln(3x+9)-7[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165135194730\">For the following exercises, state the domain, vertical asymptote, and end behavior of the function.<\/p>\n<div id=\"fs-id1165135194734\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135194736\" data-type=\"problem\">\n<p><span class=\"os-number\">16<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=\\ln(2-x)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137427662\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137427665\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137427662-solution\">17<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=\\log\\left(x-\\frac{3}{7}\\right)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137431240\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137431242\" data-type=\"problem\">\n<p><span class=\"os-number\">18<\/span><span class=\"os-divider\">. <\/span> [latex]h(x)=-\\log(3x-4)+3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134194950\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134194952\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165134194950-solution\">19<\/a><span class=\"os-divider\">. <\/span> [latex]g(x)=\\ln(2x+6)-5[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135532457\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135532459\" data-type=\"problem\">\n<p><span class=\"os-number\">20<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=\\log_3(15-5x)+6[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165137426354\">For the following exercises, state the domain, range, and <em data-effect=\"italics\">x<\/em>&#8211; and <em data-effect=\"italics\">y<\/em>-intercepts, if they exist. If they do not exist, write DNE.<\/p>\n<div id=\"fs-id1165137426368\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137426370\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137426368-solution\">21<\/a><span class=\"os-divider\">. <\/span> [latex]h(x)=\\log_4(x-1)+1[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137755640\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135543374\" data-type=\"problem\">\n<p><span class=\"os-number\">22<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=\\log(5x+10)+3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135210139\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135210141\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135210139-solution\">23<\/a><span class=\"os-divider\">. <\/span> [latex]g(x)=\\ln(-x)-2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137810010\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137810012\" data-type=\"problem\">\n<p><span class=\"os-number\">24<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=\\log_2(x+2)-5[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137942453\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137942455\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137942453-solution\">25<\/a><span class=\"os-divider\">. <\/span> [latex]h(x)=3\\ln(x)=9[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<section id=\"fs-id1165137422264\" data-depth=\"2\">\n<h3 data-type=\"title\">Graphical<\/h3>\n<p id=\"fs-id1165137422269\">For the following exercises, match each function in Figure 17 with the letter corresponding to its graph.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_976\" aria-describedby=\"caption-attachment-976\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-976\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-17-300x271.webp\" alt=\"\" width=\"300\" height=\"271\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-17-300x271.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-17-65x59.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-17-225x203.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-17-350x316.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-17.webp 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-976\" class=\"wp-caption-text\">Figure 17<\/figcaption><\/figure>\n<div id=\"fs-id1165134032273\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134032275\" data-type=\"problem\">\n<p><span class=\"os-number\">26<\/span><span class=\"os-divider\">. <\/span> [latex]d(x)=\\log(x)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<div id=\"fs-id1165135194197\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137911558\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135194197-solution\">27<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=\\ln(x)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134255566\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134255568\" data-type=\"problem\">\n<p><span class=\"os-number\">28<\/span><span class=\"os-divider\">. <\/span> [latex]g(x)=\\log_2(x)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137758154\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137758156\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137758154-solution\">29<\/a><span class=\"os-divider\">. <\/span> [latex]h(x)=\\log_5(x)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135571665\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135571667\" data-type=\"problem\">\n<p><span class=\"os-number\">30<\/span><span class=\"os-divider\">. <\/span> [latex]j(h)=\\log_{25}(x)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"eip-312\">For the following exercises, match each function in Figure 18 with the letter corresponding to its graph.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_977\" aria-describedby=\"caption-attachment-977\" style=\"width: 233px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-977\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-18-233x300.webp\" alt=\"\" width=\"233\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-18-233x300.webp 233w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-18-65x84.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-18-225x289.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-18.webp 342w\" sizes=\"auto, (max-width: 233px) 100vw, 233px\" \/><figcaption id=\"caption-attachment-977\" class=\"wp-caption-text\">Figure 18<\/figcaption><\/figure>\n<div id=\"fs-id1165134380389\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135400154\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165134380389-solution\">31<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=\\log_{\\frac{1}{3}}(x)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135206089\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135206092\" data-type=\"problem\">\n<p><span class=\"os-number\">32<\/span><span class=\"os-divider\">. <\/span> [latex]g(x)=\\log_2(x)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137855381\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137855383\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137855381-solution\">33<\/a><span class=\"os-divider\">. <\/span> [latex]h(x)=\\log_{\\frac{3}{4}}(x)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"eip-967\">For the following exercises, sketch the graphs of each pair of functions on the same axis.<\/p>\n<div id=\"fs-id1165135394346\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135394348\" data-type=\"problem\">\n<p><span class=\"os-number\">34<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=\\log(x) \\ \\text{and } g(x)=10^x[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137862457\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137862459\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137862457-solution\">35<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=\\log(x) \\ \\text{and } g(x)=\\log_{\\frac{1}{2}}(x)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135180056\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135180058\" data-type=\"problem\">\n<p><span class=\"os-number\">36<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=\\log_4(x) \\ \\text{and } g(x)=\\ln(x)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135536589\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134325846\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135536589-solution\">37<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=e^x \\ \\text{and } g(x)=\\ln(x)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165135496626\">For the following exercises, match each function in Figure 19 with the letter corresponding to its graph.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_978\" aria-describedby=\"caption-attachment-978\" style=\"width: 298px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-978\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-19-298x300.webp\" alt=\"\" width=\"298\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-19-298x300.webp 298w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-19-150x150.webp 150w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-19-65x66.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-19-225x227.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-19-350x353.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4-fig-19.webp 374w\" sizes=\"auto, (max-width: 298px) 100vw, 298px\" \/><figcaption id=\"caption-attachment-978\" class=\"wp-caption-text\">Figure 19<\/figcaption><\/figure>\n<div id=\"fs-id1165135189876\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135189878\" data-type=\"problem\">\n<p><span class=\"os-number\">38<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=\\log_4(-x+2)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137656835\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137656837\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137656835-solution\">39<\/a><span class=\"os-divider\">. <\/span> [latex]g(x)=-\\log_4(x+2)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135632110\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135632112\" data-type=\"problem\">\n<p><span class=\"os-number\">40<\/span><span class=\"os-divider\">. <\/span> [latex]h(x)=\\log_4(x+2)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165135571794\">For the following exercises, sketch the graph of the indicated function.<\/p>\n<div id=\"fs-id1165135571798\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135571800\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135571798-solution\">41<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=\\log_2(x+2)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137849072\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137849074\" data-type=\"problem\">\n<p><span class=\"os-number\">42<\/span><span class=\"os-divider\">. <\/span> [latex]f(x)=2\\log(x)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135251375\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135251377\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135251375-solution\">43<\/a><span class=\"os-divider\">. <\/span> [latex]f(x)=\\ln(-x)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134032422\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134032425\" data-type=\"problem\">\n<p><span class=\"os-number\">44<\/span><span class=\"os-divider\">. <\/span> [latex]g(x)=\\log(4x+16)+4[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137932668\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137932670\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137932668-solution\">45<\/a><span class=\"os-divider\">. <\/span> [latex]g(x)=\\log(6-3x)+1[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137705417\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137705419\" data-type=\"problem\">\n<p><span class=\"os-number\">46<\/span><span class=\"os-divider\">. <\/span> [latex]h(x)=-\\frac{1}{2}\\ln(x+1)-3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165135439867\">For the following exercises, write a logarithmic equation corresponding to the graph shown.<\/p>\n<div id=\"fs-id1165135439871\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135439873\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135439871-solution\">47<\/a><span class=\"os-divider\">. <\/span>Use [latex]y=\\log_2(x)[\/latex] as the parent function.<\/p>\n<div class=\"os-problem-container\">\n<p><span id=\"fs-id1165135443953\" data-type=\"media\" data-alt=\"The graph y=log_2(x) has been reflected over the y-axis and shifted to the right by 1.\" data-display=\"block\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-979\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4.47-298x300.webp\" alt=\"\" width=\"298\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4.47-298x300.webp 298w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4.47-150x150.webp 150w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4.47-65x66.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4.47-225x227.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4.47-350x353.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4.47.webp 375w\" sizes=\"auto, (max-width: 298px) 100vw, 298px\" \/><br \/>\n<\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135191066\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135191068\" data-type=\"problem\">\n<p><span class=\"os-number\">48<\/span><span class=\"os-divider\">. <\/span>Use [latex]f(x)=\\log_3(x)[\/latex] as the parent function.<\/p>\n<div class=\"os-problem-container\">\n<p><span id=\"fs-id1165137674237\" data-type=\"media\" data-alt=\"The graph y=log_3(x) has been reflected over the x-axis, vertically stretched by 3, and shifted to the left by 4.\" data-display=\"block\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-980\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4.48-298x300.webp\" alt=\"\" width=\"298\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4.48-298x300.webp 298w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4.48-150x150.webp 150w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4.48-65x66.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4.48-225x227.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4.48-350x353.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4.48.webp 375w\" sizes=\"auto, (max-width: 298px) 100vw, 298px\" \/><br \/>\n<\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137674252\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137674255\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137674252-solution\">49<\/a><span class=\"os-divider\">. <\/span>Use [latex]f(x)=\\log_4(x)[\/latex] as the parent function.<\/p>\n<div class=\"os-problem-container\">\n<p><span id=\"fs-id1165134086070\" data-type=\"media\" data-alt=\"The graph y=log_4(x) has been vertically stretched by 3, and shifted to the left by 2.\" data-display=\"block\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-981\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4.49-298x300.webp\" alt=\"\" width=\"298\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4.49-298x300.webp 298w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4.49-150x150.webp 150w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4.49-65x66.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4.49-225x227.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4.49-350x353.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4.49.webp 375w\" sizes=\"auto, (max-width: 298px) 100vw, 298px\" \/><br \/>\n<\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135203463\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135203465\" data-type=\"problem\">\n<p><span class=\"os-number\">50<\/span><span class=\"os-divider\">. <\/span>Use [latex]f(x)=\\log_5(x)[\/latex] as the parent function.<\/p>\n<div class=\"os-problem-container\">\n<p><span id=\"fs-id1165137705189\" data-type=\"media\" data-alt=\"The graph y=log_3(x) has been reflected over the x-axis and y-axis, vertically stretched by 2, and shifted to the right by 5.\" data-display=\"block\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-medium wp-image-982\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4.50-298x300.webp\" alt=\"\" width=\"298\" height=\"300\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4.50-298x300.webp 298w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4.50-150x150.webp 150w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4.50-65x66.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4.50-225x227.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4.50-350x353.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.4.50.webp 375w\" sizes=\"auto, (max-width: 298px) 100vw, 298px\" \/><br \/>\n<\/span><\/p>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<section id=\"fs-id1165137705205\" data-depth=\"2\">\n<h3 data-type=\"title\">Technology<\/h3>\n<p id=\"fs-id1165137705211\">For the following exercises, use a graphing calculator to find approximate solutions to each equation.<\/p>\n<div id=\"fs-id1165137705215\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137705217\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165137705215-solution\">51<\/a><span class=\"os-divider\">. <\/span> [latex]\\log(x-1)+2=\\ln(x-1)+2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<div id=\"fs-id1165137674311\" class=\"material-set-2\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137674313\" data-type=\"problem\">\n<p><span class=\"os-number\">52<\/span><span class=\"os-divider\">. <\/span> [latex]\\log(2x-3)+2=-\\log(2x-3)+5[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135176226\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135176229\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135176226-solution\">53<\/a><span class=\"os-divider\">. <\/span> [latex]\\ln(x-2)=-\\ln(x+1)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135181791\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135181793\" data-type=\"problem\">\n<p><span class=\"os-number\">54<\/span><span class=\"os-divider\">. <\/span> [latex]2\\ln(5x+1)=\\frac{1}{2}\\ln(-5x)+1[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135367864\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135367866\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135367864-solution\">55<\/a><span class=\"os-divider\">. <\/span> [latex]\\frac{1}{3}\\log(1-x)=\\log(x+1)+\\frac{1}{3}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<section id=\"fs-id1165135485736\" data-depth=\"2\">\n<h3 data-type=\"title\">Extensions<\/h3>\n<div id=\"fs-id1165135485742\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135485744\" data-type=\"problem\">\n<p><span class=\"os-number\">56<\/span><span class=\"os-divider\">. <\/span>Let <em>b<\/em> be any positive real number such that [latex]b\\not=1.[\/latex] What must [latex]\\log_b1[\/latex] be equal to? Verify the result.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135193274\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135193276\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135193274-solution\">57<\/a><span class=\"os-divider\">. <\/span>Explore and discuss the graphs of [latex]f(x)=\\log_{\\frac{1}{2}}(x)[\/latex] and [latex]g(x)=-\\log_2(x).[\/latex] Make a conjecture based on the result.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135369592\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135369594\" data-type=\"problem\">\n<p><span class=\"os-number\">58<\/span><span class=\"os-divider\">. <\/span>Prove the conjecture made in the previous exercise.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135369601\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135369603\" data-type=\"problem\">\n<p><a class=\"os-number\" href=\"chapter-6\" data-page-slug=\"chapter-6\" data-page-uuid=\"cf8ef218-9e46-5daa-bd6e-695fe7c338bb\" data-page-fragment=\"fs-id1165135369601-solution\">59<\/a><span class=\"os-divider\">. <\/span>What is the domain of the function [latex]f(x)=\\ln\\left(\\frac{x+2}{x-4}\\right)?[\/latex] Discuss the result.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137838658\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137838660\" data-type=\"problem\">\n<p><span class=\"os-number\">60<\/span><span class=\"os-divider\">. <\/span>Use properties of exponents to find the <em data-effect=\"italics\">x<\/em>-intercepts of the function [latex]f(x)=\\log(x^2+4x+4)[\/latex] algebraically. Show the steps for solving, and then verify the result by graphing the function.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n","protected":false},"author":158,"menu_order":2,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-241","chapter","type-chapter","status-publish","hentry"],"part":160,"_links":{"self":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/241","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/users\/158"}],"version-history":[{"count":12,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/241\/revisions"}],"predecessor-version":[{"id":1653,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/241\/revisions\/1653"}],"part":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/parts\/160"}],"metadata":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/241\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/media?parent=241"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=241"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/contributor?post=241"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/license?post=241"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}