{"id":240,"date":"2025-04-09T17:33:13","date_gmt":"2025-04-09T17:33:13","guid":{"rendered":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/6-3-logarithmic-functions-college-algebra-2e-openstax\/"},"modified":"2025-08-29T17:27:57","modified_gmt":"2025-08-29T17:27:57","slug":"6-3-logarithmic-functions","status":"publish","type":"chapter","link":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/chapter\/6-3-logarithmic-functions\/","title":{"raw":"6.3 Logarithmic Functions","rendered":"6.3 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_746b4be7-5dfd-4d01-8293-06ef750e0365\" 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>Convert from logarithmic to exponential form.<\/li>\r\n \t<li>Convert from exponential to logarithmic form.<\/li>\r\n \t<li>Evaluate logarithms.<\/li>\r\n \t<li>Use common logarithms.<\/li>\r\n \t<li>Use natural logarithms.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<\/div>\r\n\r\n[caption id=\"attachment_953\" align=\"aligncenter\" width=\"300\"]<img class=\"wp-image-953 size-medium\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.3-fig-1-300x200.webp\" alt=\"\" width=\"300\" height=\"200\" \/> Figure 1. Devastation of March 11, 2011 earthquake in Honshu, Japan. (credit: Daniel Pierce)[\/caption]\r\n\r\n<\/div>\r\n<p id=\"fs-id1165137557013\" class=\"has-noteref\">In 2010, a major earthquake struck Haiti, destroying or damaging over 285,000 homes<sup id=\"footnote-ref1\" data-type=\"footnote-number\"><\/sup>. One year later, another, stronger earthquake devastated Honshu, Japan, destroying or damaging over 332,000 buildings,<sup id=\"footnote-ref2\" data-type=\"footnote-number\"><\/sup> like those shown in Figure 1. Even though both caused substantial damage, the earthquake in 2011 was 100 times stronger than the earthquake in Haiti. How do we know? The magnitudes of earthquakes are measured on a scale known as the Richter Scale. The Haitian earthquake registered a 7.0 on the Richter Scale whereas the Japanese earthquake registered a 9.0.<\/p>\r\n<p id=\"fs-id1165137760714\">The Richter Scale is a base-ten logarithmic scale. In other words, an earthquake of magnitude 8 is not twice as great as an earthquake of magnitude 4. It is [latex] 10^{8-4}=10^4=10,000 [\/latex] times as great! In this lesson, we will investigate the nature of the Richter Scale and the base-ten function upon which it depends.<\/p>\r\n\r\n<section id=\"fs-id1165137644550\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Converting from Logarithmic to Exponential Form<\/h2>\r\n<p id=\"fs-id1165135192781\">In order to analyze the magnitude of earthquakes or compare the magnitudes of two different earthquakes, we need to be able to convert between logarithmic and exponential form. For example, suppose the amount of energy released from one earthquake were 500 times greater than the amount of energy released from another. We want to calculate the difference in magnitude. The equation that represents this problem is [latex] 10^x=500, [\/latex]where <em>x<\/em> represents the difference in magnitudes on the <span id=\"term-00001\" class=\"no-emphasis\" data-type=\"term\">Richter Scale<\/span>. How would we solve for <em>x<\/em>?<\/p>\r\n<p id=\"fs-id1165135160312\">We have not yet learned a method for solving exponential equations. None of the algebraic tools discussed so far is sufficient to solve [latex] 10^x=500. [\/latex] We know that [latex] 10^2=100 [\/latex] and [latex] 10^3=1000, [\/latex] so it is clear that x must be some value between 2 and 3, since [latex] y=10^x [\/latex] is increasing. We can examine a graph, as in Figure 2, to better estimate the solution.<\/p>\r\n&nbsp;\r\n\r\n[caption id=\"attachment_955\" align=\"aligncenter\" width=\"300\"]<img class=\"size-medium wp-image-955\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.3-fig-2-300x294.webp\" alt=\"\" width=\"300\" height=\"294\" \/> Figure 2[\/caption]\r\n<p id=\"fs-id1165137662989\">Estimating from a graph, however, is imprecise. To find an algebraic solution, we must introduce a new function. Observe that the graph in Figure 2 passes the horizontal line test. The exponential function [latex] y=b^x [\/latex] is one-to-one, so its inverse, [latex] x=b^y [\/latex] is also a function. As is the case with all inverse functions, we simply interchange <em>x<\/em> and <em>y<\/em> and solve for <em>y<\/em> to find the inverse function. To represent <em>y<\/em> as a function of <em>x<\/em>, we use a logarithmic function of the form [latex] y=\\log_b(x) [\/latex] The base <em>b <\/em><strong>logarithm<\/strong> of a number is the exponent by which we must raise <em>b<\/em> to get that number.<\/p>\r\n<p id=\"fs-id1165137404844\">We read a logarithmic expression as, \u201cThe logarithm with base <em>b<\/em> of <em>x<\/em> is equal to <em>y<\/em>,\" or, simplified, \u201clog base <em>b<\/em> of <em>x<\/em> is <em>y<\/em>.\" We can also say, \u201c<em>b<\/em> raised to the power of <em>y<\/em> is <em>x<\/em>,\" because logs are exponents. For example, the base 2 logarithm of 32 is 5, because 5 is the exponent we must apply to 2 to get 32. Since [latex] 2^5=32, [\/latex] we can write [latex] \\log_232=5. [\/latex] We read this as \u201clog base 2 of 32 is 5.\u201d<\/p>\r\n<p id=\"fs-id1165137597501\">We can express the relationship between logarithmic form and its corresponding exponential form as follows:<\/p>\r\n<p style=\"text-align: center;\">[latex] \\log_b(x) = y \\Leftrightarrow b^y = x, \\quad b &gt; 0, b \\neq 1 [\/latex]<\/p>\r\n<p id=\"fs-id1165137678993\">Note that the base <em>b<\/em> is always positive.<\/p>\r\n<span id=\"fs-id1165137696233\" data-type=\"media\" data-alt=\"\" data-display=\"block\">\r\n<img class=\"size-medium wp-image-956 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.3-base-b-300x51.webp\" alt=\"\" width=\"300\" height=\"51\" \/>\r\n<\/span>\r\n<p id=\"fs-id1165137400957\">Because logarithm is a function, it is most correctly written as [latex] \\log_b(x), [\/latex] using parentheses to denote function evaluation, just as we would with [latex] f(x). [\/latex] However, when the input is a single variable or number, it is common to see the parentheses dropped and the expression written without parentheses, as [latex] \\log_bx. [\/latex] Note that many calculators require parentheses around the <em>x<\/em>.<\/p>\r\n<p id=\"fs-id1165137827516\">We can illustrate the notation of logarithms as follows:<\/p>\r\n<span id=\"fs-id1165137771679\" data-type=\"media\" data-alt=\"\" data-display=\"block\">\r\n<img class=\"size-medium wp-image-957 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.3-log-c-300x62.webp\" alt=\"\" width=\"300\" height=\"62\" \/>\r\n<\/span>\r\n<p id=\"fs-id1165137575165\">Notice that, comparing the logarithm function and the exponential function, the input and the output are switched. This means [latex] y=\\log_b(x) [\/latex] and [latex] y=b^x [\/latex] are inverse functions.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Definition of the Logarithmic Function<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nA <strong>logarithm <\/strong>base <em>b<\/em> of a positive number <em>x <\/em>satisfies the following definition.\r\n\r\nFor [latex] x&gt; 0, b&gt; 0, b\\not=1, [\/latex]\r\n<p style=\"text-align: center;\">[latex] y=\\log_b(x) [\/latex] is\u00a0equivalent\u00a0to [latex] b_y=x [\/latex]<\/p>\r\nwhere,\r\n<ul>\r\n \t<li>we read [latex] \\log_b(x) [\/latex] as, \u201cthe logarithm with base <em>b <\/em>of <em>x<\/em>\" or the \u201clog base <em>b<\/em> of <em>x<\/em>.\"<\/li>\r\n \t<li>the logarithm <em>y<\/em> is the exponent to which <em>b<\/em> must be raised to get <em>x.<\/em><\/li>\r\n<\/ul>\r\nAlso, since the logarithmic and exponential functions switch the <em>x<\/em> and <em>y<\/em> values, the domain and range of the exponential function are interchanged for the logarithmic function. Therefore,\r\n<ul>\r\n \t<li>the domain of the logarithm function with base <em>b<\/em> is [latex] (0, \\infty). [\/latex]<\/li>\r\n \t<li>the range of the logarithm function with base <em>b<\/em> is [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\">Q&amp;A<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\n<strong>Q: Can we take the logarithm of a negative number?<\/strong>\r\n\r\n<em data-effect=\"italics\">A: No. Because the base of an exponential function is always positive, no power of that base can ever be negative. We can never take the logarithm of a negative number. Also, we cannot take the logarithm of zero. Calculators may output a log of a negative number when in complex mode, but the log of a negative number is not a real number.<\/em>\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 an equation in logarithmic form [latex] lob_b(x)=y, [\/latex] convert it to exponential form. <\/strong>\r\n<ol>\r\n \t<li>Examine the equation [latex] \\log_b(x), [\/latex] and identify b, y, and x.<\/li>\r\n \t<li>Rewrite [latex] \\log_b(x)=y [\/latex] as [latex] b^y=x. [\/latex]<\/li>\r\n<\/ol>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 1: Converting from Logarithmic Form to Exponential Form<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWrite the following logarithmic equations in exponential form.\r\n\r\n(a) [latex] \\log_6(\\sqrt{6})=\\frac{1}{2} [\/latex]\r\n\r\n(b) [latex] \\log_3(9)=2 [\/latex]\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>First, identify the values of <em>b<\/em>, <em>y<\/em>, and <em>x<\/em>. Then, write the equation in the form [latex] b^y=x. [\/latex]\r\n\r\n(a) [latex] \\log_6(\\sqrt{6})=\\frac{1}{2} [\/latex]\r\n\r\nHere, [latex] b=6, y=\\frac{1}{2}, [\/latex] and [latex] x=\\sqrt{6}. [\/latex] Therefore, the equation [latex] \\log_6(\\sqrt{6})=\\frac{1}{2} [\/latex] is equivalent to [latex] 6^{\\frac{1}{2}}=\\sqrt{6}. [\/latex]\r\n\r\n(b) [latex] \\log_3(9)=2 [\/latex]\r\n\r\nHere, [latex] b=3, y=2, [\/latex] and [latex] x=9. [\/latex] Therefore, the equation [latex] \\log_3(9)=2 [\/latex] is equivalent to [latex] 3^2=9. [\/latex]\r\n\r\n<\/details><\/div>\r\n<\/div>\r\n<section><\/section>\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\nWrite the following logarithmic equations in exponential form.\r\n\r\n(a) [latex] \\log_{10}(1,000,000)=6 [\/latex]\r\n\r\n(b) [latex] \\log_5(25)=2 [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<section id=\"fs-id1165137585244\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Converting from Exponential to Logarithmic Form<\/h2>\r\n<p id=\"fs-id1165137933968\">To convert from exponents to logarithms, we follow the same steps in reverse. We identify the base <em>b<\/em>, exponent\u00a0<em>x<\/em>, and output <em>y<\/em>. Then we write [latex] x=\\log^b(y). [\/latex]<\/p>\r\n\r\n<div class=\"textbox textbox--exercises\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Example 2: Converting from Exponential Form to Logarithmic Form<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWrite the following exponential equations in logarithmic form.\r\n\r\n(a) [latex] 2^3=8 [\/latex]\r\n\r\n(b) [latex] 5^2=25 [\/latex]\r\n\r\n(c) [latex] 10^{-4}=\\frac{1}{10,000} [\/latex]\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>First, identify the values of <em>b<\/em>, <em>y<\/em>, and <em>x<\/em>. Then, write the equation in the form [latex] x=\\log_b(y). [\/latex]\r\n\r\n(a) [latex] 2^3=8 [\/latex]\r\n\r\nHere, [latex] b=2, x=3, [\/latex] and [latex] y=8. [\/latex] Therefore, the equation [latex] 2^3=8 [\/latex] is equivalent to [latex] \\log_2(8)=3. [\/latex]\r\n\r\n(b) [latex] 5^5=25 [\/latex]\r\n\r\nHere, [latex] b=5, x=2, [\/latex] and [latex] y=25. [\/latex] Therefore, the equation [latex] 5^2=25 [\/latex] is equivalent to [latex] \\log_5(25)=2. [\/latex]\r\n\r\n(c) [latex] 10^{-4}=\\frac{1}{10,000} [\/latex]\r\n\r\nHere, [latex] b=10, x=-4, [\/latex] and [latex] y=\\frac{1}{10,000}. [\/latex] Therefore, the equation [latex] 10^{-4}=\\frac{1}{10,000} [\/latex] is equivalent to [latex] \\log_{10}\\left(\\frac{1}{10,000}\\right)=-4. [\/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 #2<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nWrite the following exponential equations in logarithmic form.\r\n\r\n(a) [latex] 3^2=9 [\/latex]\r\n\r\n(b) [latex] 5^3=125 [\/latex]\r\n\r\n(c) [latex] 2^{-1}=\\frac{1}{2} [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1165137530906\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Evaluating Logarithms<\/h2>\r\n<p id=\"fs-id1165137422589\">Knowing the squares, cubes, and roots of numbers allows us to evaluate many logarithms mentally. For example, consider [latex] \\log_28. [\/latex] We ask, \u201cTo what exponent must 2 be raised in order to get 8?\u201d Because we already know [latex] 2^3=8, [\/latex] it follows that [latex] log_28=3. [\/latex]<\/p>\r\n<p id=\"fs-id1165137733822\">Now consider solving [latex] \\log_749 [\/latex] and [latex] \\log_327 [\/latex] mentally.<\/p>\r\n\r\n<ul id=\"fs-id1165137937690\">\r\n \t<li>We ask, \u201cTo what exponent must 7 be raised in order to get 49?\u201d We know [latex] 7^2=49. [\/latex] Therefore, [latex] \\log_749=2. [\/latex]<\/li>\r\n \t<li>We ask, \u201cTo what exponent must 3 be raised in order to get 27?\u201d We know [latex] 3^3=27. [\/latex] Therefore, [latex] \\log_327=3. [\/latex]<\/li>\r\n<\/ul>\r\n<p id=\"fs-id1165137456358\">Even some seemingly more complicated logarithms can be evaluated without a calculator. For example, let\u2019s evaluate [latex] \\log_{\\frac{2}{3}}\\frac{4}{9} [\/latex] mentally.<\/p>\r\n\r\n<ul id=\"fs-id1165137584208\">\r\n \t<li>We ask, \u201cTo what exponent must [latex] \\frac{2}{3} [\/latex] be raised in order to get [latex] \\frac{4}{9}?\" [\/latex] We know [latex] 2^2=4 [\/latex] and [latex] 3^2=9, [\/latex] so [latex] \\left(\\frac{2}{3}\\right)^2=\\frac{4}{9}. [\/latex] Therefore, [latex] \\log_{\\frac{2}{3}}\\left(\\frac{4}{9}\\right)=2. [\/latex]<\/li>\r\n<\/ul>\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 logarithm of the form [latex] y=\\log^b(x), [\/latex] evaluate it mentally.<\/strong>\r\n<ol>\r\n \t<li>Rewrite the argument <em>x<\/em> as a power of [latex]b: b^y=x. [\/latex]<\/li>\r\n \t<li>Use previous knowledge of powers of <em>b<\/em> identify <em>y<\/em> by asking, \u201cTo what exponent should b be raised in order to get x?\"<\/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 3: Solving Logarithms Mentally<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nSolve [latex] y=\\log_4(64) [\/latex] without using a calculator.\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>First we rewrite the logarithm in exponential form: [latex] 4^y=64. [\/latex] Next, we ask, \u201cTo what exponent must 4 be raised in order to get 64?\u201d\r\n\r\nWe know\r\n<p style=\"text-align: center;\">[latex] 4^3=64 [\/latex]<\/p>\r\nTherefore,\r\n<p style=\"text-align: center;\">[latex] \\log_4(64)=3 [\/latex]<\/p>\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\nSolve [latex] y=\\log_{121}(11) [\/latex] without using a calculator.\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 4: Evaluating the Logarithm of a Reciprocal<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nEvaluate [latex] y=\\log_3\\left(\\frac{1}{27}\\right) [\/latex] without using a calculator.\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>First we rewrite the logarithm in exponential form: [latex] 3^y=\\frac{1}{27}. [\/latex] Next, we ask, \u201cTo what exponent must 3 be raised in order to get [latex] \\frac{1}{27}?\" [\/latex]\r\n\r\nWe know [latex] 3^3=27, [\/latex] but what must we do to get the reciprocal, [latex] \\frac{1}{27}? [\/latex] Recall from working with exponents that [latex] b^{-a}=\\frac{1}{b^a}. [\/latex] We use this information to write\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{rl} 3^{-3} &amp;=&amp; \\frac{1}{3^3} \\\\ &amp;=&amp; \\frac{1}{27} \\end{array} [\/latex]<\/p>\r\nTherefore, [latex] \\log_3\\left(\\frac{1}{27}\\right)=-3. [\/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 #4<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nEvaluate [latex] y=\\log_2\\left(\\frac{1}{32}\\right) [\/latex] without using a calculator.\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1165137547253\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Using Common Logarithms<\/h2>\r\n<p id=\"fs-id1165137574205\">Sometimes you may see a logarithm written without a base. When you see one written this way, you need to look at the expression before evaluating it. It may be that the base you use doesn't matter. If you find it in computer science, it often means [latex] \\log_2(x). [\/latex] However, in mathematics it almost always means the common logarithm of 10. In other words, the expression [latex] \\log(x) [\/latex] often means [latex] \\log_{10}(x). [\/latex]<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Definition of the Common Logarithm<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nA <strong>common logarithm<\/strong> is a logarithm with base 10. We can also write [latex] \\log_{10}(x) [\/latex] simply as [latex] \\log(x). [\/latex] The common logarithm of a positive number <em>x<\/em> satisfies the following definition.\r\n\r\nFor [latex] x&gt; 0, [\/latex]\r\n<p style=\"text-align: center;\">[latex] y=\\log(x) [\/latex] is\u00a0equivalent\u00a0to [latex] y=\\log_{10}(x) [\/latex]<\/p>\r\nWe read [latex] log(x) [\/latex] as, \u201cthe logarithm with base 10 of <em>x<\/em>\" or \u201clog base 10 of <em>x<\/em>.\"\r\n\r\nThe logarithm <em>y<\/em> is the exponent to which 10 must be raised to get <em>x<\/em>.\r\n\r\n<\/div>\r\n<\/div>\r\n<p id=\"para-00002\">Currently, we use [latex] \\log_b(x), lg(x) [\/latex] as the common logarithm, [latex] \\operatorname{lb}(x)\u00a0[\/latex] as the binary logarithm, and [latex] \\ln(x) [\/latex] as the natural logarithm. Writing [latex] \\lg(x) [\/latex] without specifying a base is now considered bad form, despite being frequently found in older materials.<\/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 common logarithm of the form [latex] y=\\log(x), [\/latex] evaluate it mentally. <\/strong>\r\n<ol>\r\n \t<li>Rewrite the argument <em>x<\/em> as a power of [latex] 10: 10^y=x. [\/latex]<\/li>\r\n \t<li>Use previous knowledge of powers of 10 to identify <em>y<\/em> by asking, \u201cTo what exponent must 10 be raised in order to get <em>x<\/em>?\"<\/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: Finding the Value of a Common Logarithm Mentally<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nEvaluate [latex] y=\\log(1000) [\/latex] without using a calculator.\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>First we rewrite the logarithm in exponential form: [latex] 10^y=1000. [\/latex] Next, we ask, \u201cTo what exponent must 10 be raised in order to get 1000?\u201d We know\r\n<p style=\"text-align: center;\">[latex] 10^3=1000 [\/latex]<\/p>\r\nTherefore, [latex] \\log(1000)=3. [\/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\nEvaluate [latex] y=\\log(1,000,000) [\/latex]\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 common logarithm with the form [latex] y=\\log(x), [\/latex] evaluate it using a calculator. <\/strong>\r\n<ol>\r\n \t<li>Press <strong>[LOG]<\/strong>.<\/li>\r\n \t<li>Enter the value given for <em>x<\/em> followed by <strong>[ ) ]<\/strong>.<\/li>\r\n \t<li>Press <strong>[ENTER]<\/strong>.<\/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: Finding the Value of a Common Logarithm Using a Calculator<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nEvaluate [latex] y=\\log(321) [\/latex] to four decimal places using a calculator.\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>\r\n<ul>\r\n \t<li>Press <strong>[LOG<\/strong><\/li>\r\n \t<li>Enter 321, followed by <strong>[ ) ]<\/strong>.<\/li>\r\n \t<li>Press <strong>[ENTER]<\/strong>.<\/li>\r\n<\/ul>\r\nRounding to four decimal places, [latex] \\log(321)\\approx 2.5065. [\/latex]\r\n<h3>Analysis<\/h3>\r\nNote that [latex] 10^2=100 [\/latex] and that [latex] 10^3=1000. [\/latex] Since 321 is between 100 and 1000, we know that [latex] \\log(321) [\/latex] must be between [latex] \\log(100) [\/latex] and [latex] \\log(1000). [\/latex] This gives us the following:\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{ccc} 1000 &amp;&lt;&amp; 321 &amp;&lt;&amp; 1000 \\\\ 2 &amp;&lt;&amp; 2.5065 &amp;&lt;&amp; 3 \\end{array} [\/latex]<\/p>\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\nEvaluate [latex] y=\\log(123) [\/latex] to four decimal places using a calculator.\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: Rewriting and Solving a Real-World Exponential Model<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nThe amount of energy released from one earthquake was 500 times greater than the amount of energy released from another. The equation [latex] 10^x=500 [\/latex] represents this situation, where x is the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes?\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>We begin by rewriting the exponential equation in logarithmic form.\r\n<p style=\"text-align: center;\">[latex] \\begin{array}{lll} 10^x &amp;=&amp; 500 \\\\ \\log(500) &amp;=&amp; x &amp;&amp; \\text{Use the definition of the common log.} \\end{array} [\/latex]<\/p>\r\nNext we evaluate the logarithm using a calculator:\r\n<ul>\r\n \t<li>Press <strong>[LOG]<\/strong>.<\/li>\r\n \t<li>Enter 500, followed by <strong>[ ) ]<\/strong>.<\/li>\r\n \t<li>Press <strong>[ENTER]<\/strong>.<\/li>\r\n \t<li>To the nearest thousandth, [latex] \\log(500)\\approx 2.699. [\/latex]<\/li>\r\n<\/ul>\r\nThe difference in magnitudes was about [latex] 2.699. [\/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\nThe amount of energy released from one earthquake was 8,500 times greater than the amount of energy released from another. The equation [latex] 10^x=8500 [\/latex] represents this situation, where <em>x<\/em> is the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes?\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><section id=\"fs-id1165137405741\" data-depth=\"1\">\r\n<h2 data-type=\"title\">Using Natural Logarithms<\/h2>\r\n<p id=\"fs-id1165137661970\">The most frequently used base for logarithms is <em>e<\/em>, the value of which is approximately [latex] 2.71828. [\/latex] Base <em>e<\/em> logarithms are important in calculus and some scientific applications; they are called <strong>natural logarithms<\/strong>. The base <em>e<\/em> logarithm, [latex] log_e(x), [\/latex] has its own notation, [latex] ln(x). [\/latex]<\/p>\r\n<p id=\"fs-id1165137473872\">Most values of [latex] ln(x) [\/latex] can be found only using a calculator. The major exception is that, because the logarithm of 1 is always 0 in any base, [latex] \\ln1=0. [\/latex] For other natural logarithms, we can use the [latex] \\ln [\/latex] key that can be found on most scientific calculators. We can also find the natural logarithm of any power of <em>e<\/em> using the inverse property of logarithms.<\/p>\r\n\r\n<div class=\"textbox textbox--examples\"><header class=\"textbox__header\">\r\n<p class=\"textbox__title\">Definition of the Natural Logarithm<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nA <strong>natural logarithm<\/strong> is a logarithm with base <em>e<\/em>. We write [latex] \\log_e(x) [\/latex] simply as [latex] \\ln(x). [\/latex] The natural logarithm of a positive number <em>x<\/em> satisfies the following definition.\r\n\r\nFor [latex] x&gt; 0, [\/latex]\r\n<p style=\"text-align: center;\">[latex] y=\\ln(x) [\/latex] is\u00a0equivalent\u00a0to [latex] e^y=x. [\/latex]<\/p>\r\nWe read [latex] \\ln(x) [\/latex] as, \u201cthe logarithm with base <em>e<\/em> of <em>x<\/em>\u201d or \u201cthe natural logarithm of <em>x<\/em>.\u201d\r\n\r\nThe logarithm <em>y<\/em> is the exponent to which <em>e<\/em> must be raised to get <em>x<\/em>.\r\n\r\nSince the functions [latex] y=e^x [\/latex] and [latex] y=\\ln(x) [\/latex] are inverse functions, [latex] \\ln(e^x)=x [\/latex] for all x and [latex] e^{\\ln(x)=x}=x [\/latex] for [latex] x&gt; 0. [\/latex]\r\n\r\n<\/div>\r\n<\/div>\r\n<div class=\"os-note-body\">\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 natural logarithm with the form [latex] y=\\ln(x), [\/latex] evaluate it using a calculator. <\/strong>\r\n<ol>\r\n \t<li>Press <strong>[LN]<\/strong>.<\/li>\r\n \t<li>Enter the value given for <em>x<\/em>, followed by <strong>[ ) ]<\/strong>.<\/li>\r\n \t<li>Press <strong>[ENTER]<\/strong>.<\/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 8: Evaluating a Natural Logarithm Using a Calculator<\/p>\r\n\r\n<\/header>\r\n<div class=\"textbox__content\">\r\n\r\nEvaluate [latex] y=\\ln(500) [\/latex] to four decimal places using a calculator.\r\n\r\n&nbsp;\r\n\r\n<details><summary><strong>Solution (click to expand)<\/strong><\/summary>\r\n<ul>\r\n \t<li>Press <strong>[LN]<\/strong>.<\/li>\r\n \t<li>Enter 500, followed by <strong>[ ) ]<\/strong>.<\/li>\r\n \t<li>Press <strong>[ENTER]<\/strong>.<\/li>\r\n<\/ul>\r\nRounding to four decimal places, [latex] \\ln(500)\\approx 6.2146. [\/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\nEvaluate [latex] \\ln(-500). [\/latex]\r\n\r\n<\/div>\r\n<\/div>\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 this online resource for additional instruction and practice with logarithms.\r\n<ul>\r\n \t<li><a href=\"https:\/\/www.youtube.com\/watch?v=z296tOPj0HA\">Introduction to Logarithms<\/a><\/li>\r\n<\/ul>\r\n<\/div>\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\">6.3 Section Exercises<\/span><\/h2>\r\n<section id=\"fs-id1165135192789\" class=\"section-exercises\" data-depth=\"1\"><section id=\"fs-id1165137427076\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Verbal<\/h3>\r\n<div id=\"fs-id1165137817361\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137559978\" 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-id1165137817361-solution\">1<\/a><span class=\"os-divider\">. <\/span>What is a base b logarithm? Discuss the meaning by interpreting each part of the equivalent equations [latex] b^y=x [\/latex] and [latex] \\log_bx=y [\/latex] for [latex] b&gt; 0, b\\not=1. [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/section><\/div>\r\n<div id=\"fs-id1165137574896\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137658231\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span>How is the logarithmic function [latex] f(x)=\\log_bx [\/latex] related to the exponential function [latex] g(x)=b^x? [\/latex] What is the result of composing these two functions?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137446568\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137602953\" 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-id1165137446568-solution\">3<\/a><span class=\"os-divider\">. <\/span>How can the logarithmic equation [latex] \\log_bx=y [\/latex] be solved for x using the properties of exponents?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137470358\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135526986\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">4<\/span><span class=\"os-divider\">.\u00a0<\/span>Discuss the meaning of the common logarithm. What is its relationship to a logarithm with base <em>b <\/em>and how does the notation differ?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137507578\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137828407\" 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-id1165137507578-solution\">5<\/a><span class=\"os-divider\">. <\/span>Discuss the meaning of the natural logarithm. What is its relationship to a logarithm with base <em>b<\/em> and how does the notation differ?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<section id=\"fs-id1165137447239\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Algebraic<\/h3>\r\n<p id=\"fs-id1165137414571\">For the following exercises, rewrite each equation in exponential form.<\/p>\r\n\r\n<div id=\"fs-id1165137646887\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137664870\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">6<\/span><span class=\"os-divider\">. <\/span> [latex] \\log_4(q)=m [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section>\r\n<div id=\"fs-id1165137473551\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137454663\" 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-id1165137473551-solution\">7<\/a><span class=\"os-divider\">. <\/span> [latex] \\log_a(b)=c [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137506749\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135536326\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">8<\/span><span class=\"os-divider\">. <\/span> [latex] \\log_{16}(y)=x [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137410961\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137602136\" 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-id1165137410961-solution\">9<\/a><span class=\"os-divider\">. <\/span> [latex] \\log_x(64)=y [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137673422\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135585639\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">10<\/span><span class=\"os-divider\">. <\/span> [latex] \\log_y(x)=-11 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135356538\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137501385\" 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-id1165135356538-solution\">11<\/a><span class=\"os-divider\">. <\/span> [latex] \\log_{15}(a)=b [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137580855\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137438413\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">12<\/span><span class=\"os-divider\">. <\/span> [latex] \\log_y(137)=x [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137611652\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134038222\" 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-id1165137611652-solution\">13<\/a><span class=\"os-divider\">. <\/span> [latex] \\log_{13}(142)=a [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137724172\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137724174\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">14<\/span><span class=\"os-divider\">. <\/span> [latex] \\log(v)=t [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137735890\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137735892\" 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-id1165137735890-solution\">15<\/a><span class=\"os-divider\">. <\/span> [latex] \\ln(w)=n [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165137810335\">For the following exercises, rewrite each equation in logarithmic form.<\/p>\r\n\r\n<div id=\"fs-id1165135186264\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135203858\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">16<\/span><span class=\"os-divider\">. <\/span> [latex] 4^x=y [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135194695\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137724816\" 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-id1165135194695-solution\">17<\/a><span class=\"os-divider\">. <\/span> [latex] c^d=k [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137698394\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137434093\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">18<\/span><span class=\"os-divider\">. <\/span> [latex] m^{-7}=n [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135384408\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135384410\" 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-id1165135384408-solution\">19<\/a><span class=\"os-divider\">. <\/span> [latex] 19^x=y [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135358961\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137627327\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">20<\/span><span class=\"os-divider\">. <\/span> [latex] x^{=\\frac{10}{13}}=y [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165132971714\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165132971716\" 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-id1165132971714-solution\">21<\/a><span class=\"os-divider\">. <\/span> [latex] n^4=103 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137564489\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137564491\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">22<\/span><span class=\"os-divider\">. <\/span> [latex] \\left(\\frac{7}{5}\\right)^m=n [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137731983\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137597971\" 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-id1165137731983-solution\">23<\/a><span class=\"os-divider\">. <\/span> [latex] y^x=\\frac{39}{100} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137827739\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137762434\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">24<\/span><span class=\"os-divider\">. <\/span> [latex] 10^a=b [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135296148\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165134040600\" 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-id1165135296148-solution\">25<\/a><span class=\"os-divider\">. <\/span> [latex] e^k=h [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165137762947\">For the following exercises, solve for <em>x<\/em> by converting the logarithmic equation to exponential form.<\/p>\r\n\r\n<div id=\"fs-id1165137438777\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137438779\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">26<\/span><span class=\"os-divider\">. <\/span> [latex] \\log_3(x)=2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137482919\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137628211\" 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-id1165137482919-solution\">27<\/a><span class=\"os-divider\">. <\/span> [latex] \\log_2(x)=-3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137726345\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137726348\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">28<\/span><span class=\"os-divider\">. <\/span> [latex] \\log_5(x)=2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137389073\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137639044\" 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-id1165137389073-solution\">29<\/a><span class=\"os-divider\">. <\/span> [latex] \\log_3(x)=3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137661950\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137726952\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">30<\/span><span class=\"os-divider\">. <\/span> [latex] \\log_2(x)=6 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137627690\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137657454\" 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-id1165137627690-solution\">31<\/a><span class=\"os-divider\">. <\/span> [latex] \\log_9(x)=\\frac{1}{2} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135203739\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135203742\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">32<\/span><span class=\"os-divider\">. <\/span> [latex] \\log_{18}(x)=2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135613168\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137414109\" 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-id1165135613168-solution\">33<\/a><span class=\"os-divider\">. <\/span> [latex] \\log_6(x)=-3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137937666\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137937668\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">34<\/span><span class=\"os-divider\">. <\/span> [latex] \\log(x)=3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137736080\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137442469\" 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-id1165137736080-solution\">35<\/a><span class=\"os-divider\">. <\/span> [latex] \\ln(x)=2 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165137634430\">For the following exercises, use the definition of common and natural logarithms to simplify.<\/p>\r\n\r\n<div id=\"fs-id1165137634434\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137732545\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">36<\/span><span class=\"os-divider\">. <\/span> [latex] \\log(100^8) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137936925\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137936927\" 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-id1165137936925-solution\">37<\/a><span class=\"os-divider\">. <\/span> [latex] 10^{\\log(32)} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137529000\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137529002\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">38<\/span><span class=\"os-divider\">. <\/span> [latex] 32\\log(.0001) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165134079645\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137566210\" 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-id1165134079645-solution\">39<\/a><span class=\"os-divider\">. <\/span> [latex] e^{\\ln(1.06)} [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137400162\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137400164\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">40<\/span><span class=\"os-divider\">. <\/span> [latex] \\ln(e^{-5.03}) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137543862\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137543865\" 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-id1165137543862-solution\">41<\/a><span class=\"os-divider\">. <\/span> [latex] e^{\\ln(10.125)}+4 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<section id=\"fs-id1165137451809\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Numeric<\/h3>\r\n<p id=\"fs-id1165137564770\">For the following exercises, evaluate the base <em>b<\/em> logarithmic expression without using a calculator.<\/p>\r\n\r\n<div id=\"fs-id1165137676437\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137676439\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">42<\/span><span class=\"os-divider\">. <\/span> [latex] \\log_3\\left(\\frac{1}{27}\\right) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section>\r\n<div id=\"fs-id1165137590077\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137590079\" 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-id1165137590077-solution\">43<\/a><span class=\"os-divider\">. <\/span> [latex] \\log_6(\\sqrt{6}) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137415151\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137423807\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">44<\/span><span class=\"os-divider\">. <\/span> [latex] \\log_2\\left(\\frac{1}{8}\\right)+4 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135190140\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137806053\" 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-id1165135190140-solution\">45<\/a><span class=\"os-divider\">. <\/span> [latex] 6\\log_8(4) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165137532658\">For the following exercises, evaluate the common logarithmic expression without using a calculator.<\/p>\r\n\r\n<div id=\"fs-id1165137542512\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135613435\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">46<\/span><span class=\"os-divider\">. <\/span> [latex] \\log(10,000) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137755960\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137736340\" 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-id1165137755960-solution\">47<\/a><span class=\"os-divider\">. <\/span> [latex] \\log(0.001) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135190197\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135190199\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">48<\/span><span class=\"os-divider\">. <\/span> [latex] \\log(1)+7 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137432736\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137655695\" 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-id1165137432736-solution\">49<\/a><span class=\"os-divider\">. <\/span> [latex] 2\\log(100^{-3}) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<p id=\"fs-id1165137570442\">For the following exercises, evaluate the natural logarithmic expression without using a calculator.<\/p>\r\n\r\n<div id=\"fs-id1165137619029\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137619032\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">50<\/span><span class=\"os-divider\">. <\/span> [latex] \\ln(e^{\\frac{1}{3}}) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137736480\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137736482\" 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-id1165137736480-solution\">51<\/a><span class=\"os-divider\">. <\/span> [latex] \\ln(1) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137804477\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137474568\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">52<\/span><span class=\"os-divider\">. <\/span> [latex] \\ln(e^{-0.225})-3 [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137762957\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135570229\" 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-id1165137762957-solution\">53<\/a><span class=\"os-divider\">. <\/span> [latex] 25\\ln(e^{\\frac{2}{5}}) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<section id=\"fs-id1165137527430\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Technology<\/h3>\r\n<p id=\"fs-id1165137401108\">For the following exercises, evaluate each expression using a calculator. Round to the nearest thousandth.<\/p>\r\n\r\n<div id=\"fs-id1165137626617\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137626619\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">54<\/span><span class=\"os-divider\">. <\/span> [latex] \\log(0.04) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section>\r\n<div id=\"fs-id1165137640432\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137640434\" 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-id1165137640432-solution\">55<\/a><span class=\"os-divider\">. <\/span> [latex] \\ln(15) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137731720\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137731722\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">56<\/span><span class=\"os-divider\">. <\/span> [latex] \\ln\\left(\\frac{4}{5}\\right) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137527828\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137527830\" 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-id1165137527828-solution\">57<\/a><span class=\"os-divider\">. <\/span> [latex] \\log(\\sqrt{2}) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137655087\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137655089\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">58<\/span><span class=\"os-divider\">. <\/span> [latex] \\ln(\\sqrt{2}) [\/latex]\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<section id=\"fs-id1165137932472\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Extensions<\/h3>\r\n<div id=\"fs-id1165137435099\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137593336\" 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-id1165137435099-solution\">59<\/a><span class=\"os-divider\">. <\/span>Is [latex] x=0 [\/latex] in the domain of the function [latex] f(x)=\\log(x)? [\/latex] If so, what is the value of the function when [latex] x=0? [\/latex] Verify the result.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section>\r\n<div id=\"fs-id1165137597455\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137597458\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">60<\/span><span class=\"os-divider\">. <\/span>Is [latex] f(x)=0 [\/latex] in the range of the function [latex] f(x)=\\log(x)? [\/latex] If so, for what value of <em>x<\/em>? Verify the result.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137548869\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137548871\" 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-id1165137548869-solution\">61<\/a><span class=\"os-divider\">. <\/span>Is there a number <em>x<\/em> such that [latex] \\lnx=2? [\/latex] If so, what is that number? Verify the result.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137640809\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137603717\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">62<\/span><span class=\"os-divider\">. <\/span>Is the following true: [latex] \\frac{\\log_3(27)}{log_4\\left(\\frac{1}{64}\\right)}=-1? [\/latex] Verify the result.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165137453751\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137453754\" 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-id1165137453751-solution\">63<\/a><span class=\"os-divider\">. <\/span>Is the following true: [latex] \\frac{\\ln(e^{1.725})}{ln(1)}=1.725? [\/latex] Verify the result.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<section id=\"fs-id1165135437184\" data-depth=\"2\">\r\n<h3 data-type=\"title\">Real-World Applications<\/h3>\r\n<div id=\"fs-id1165137732791\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137913993\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">64<\/span><span class=\"os-divider\">. <\/span>The exposure index [latex] EI [\/latex] for a camera is a measurement of the amount of light that hits the image receptor. It is determined by the equation [latex] EI=\\log_2\\left(\\frac{f^2}{t}\\right), [\/latex] where f is the \u201cf-stop\u201d setting on the camera, and t is the exposure time in seconds. Suppose the f-stop setting is 8 and the desired exposure time is 2 seconds. What will the resulting exposure index be?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section>\r\n<div id=\"fs-id1165137602359\" class=\"os-hasSolution\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165137838750\" 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-id1165137602359-solution\">65<\/a><span class=\"os-divider\">. <\/span>Refer to the previous exercise. Suppose the light meter on a camera indicates an [latex] EI [\/latex] of -2, and the desired exposure time is 16 seconds. What should the f-stop setting be?\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<div id=\"fs-id1165135192763\" data-type=\"exercise\"><header><\/header><section>\r\n<div id=\"fs-id1165135192765\" data-type=\"problem\">\r\n\r\n<span class=\"os-number\">66<\/span><span class=\"os-divider\">. <\/span>The intensity levels <em data-effect=\"italics\">I<\/em> of two earthquakes measured on a seismograph can be compared by the formula [latex] \\log\\frac{I_1}{I_2}=M_1-M_2 [\/latex] where [latex] M [\/latex] is the magnitude given by the Richter Scale. In August 2009, an earthquake of magnitude 6.1 hit Honshu, Japan. In March 2011, that same region experienced yet another, more devastating earthquake, this time with a magnitude of 9.0.<sup id=\"footnote-ref5\" data-type=\"footnote-number\"><\/sup> How many times greater was the intensity of the 2011 earthquake? Round to the nearest whole number.\r\n\r\n<\/div>\r\n<\/section><\/div>","rendered":"<div id=\"main-content\" class=\"MainContent__ContentStyles-sc-6yy1if-0 NnXKu\" tabindex=\"-1\" data-dynamic-style=\"true\">\n<div id=\"page_746b4be7-5dfd-4d01-8293-06ef750e0365\" 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>Convert from logarithmic to exponential form.<\/li>\n<li>Convert from exponential to logarithmic form.<\/li>\n<li>Evaluate logarithms.<\/li>\n<li>Use common logarithms.<\/li>\n<li>Use natural logarithms.<\/li>\n<\/ul>\n<\/div>\n<\/div>\n<figure id=\"attachment_953\" aria-describedby=\"caption-attachment-953\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"wp-image-953 size-medium\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.3-fig-1-300x200.webp\" alt=\"\" width=\"300\" height=\"200\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.3-fig-1-300x200.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.3-fig-1-65x43.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.3-fig-1-225x150.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.3-fig-1-350x233.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.3-fig-1.webp 488w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-953\" class=\"wp-caption-text\">Figure 1. Devastation of March 11, 2011 earthquake in Honshu, Japan. (credit: Daniel Pierce)<\/figcaption><\/figure>\n<\/div>\n<p id=\"fs-id1165137557013\" class=\"has-noteref\">In 2010, a major earthquake struck Haiti, destroying or damaging over 285,000 homes<sup id=\"footnote-ref1\" data-type=\"footnote-number\"><\/sup>. One year later, another, stronger earthquake devastated Honshu, Japan, destroying or damaging over 332,000 buildings,<sup id=\"footnote-ref2\" data-type=\"footnote-number\"><\/sup> like those shown in Figure 1. Even though both caused substantial damage, the earthquake in 2011 was 100 times stronger than the earthquake in Haiti. How do we know? The magnitudes of earthquakes are measured on a scale known as the Richter Scale. The Haitian earthquake registered a 7.0 on the Richter Scale whereas the Japanese earthquake registered a 9.0.<\/p>\n<p id=\"fs-id1165137760714\">The Richter Scale is a base-ten logarithmic scale. In other words, an earthquake of magnitude 8 is not twice as great as an earthquake of magnitude 4. It is [latex]10^{8-4}=10^4=10,000[\/latex] times as great! In this lesson, we will investigate the nature of the Richter Scale and the base-ten function upon which it depends.<\/p>\n<section id=\"fs-id1165137644550\" data-depth=\"1\">\n<h2 data-type=\"title\">Converting from Logarithmic to Exponential Form<\/h2>\n<p id=\"fs-id1165135192781\">In order to analyze the magnitude of earthquakes or compare the magnitudes of two different earthquakes, we need to be able to convert between logarithmic and exponential form. For example, suppose the amount of energy released from one earthquake were 500 times greater than the amount of energy released from another. We want to calculate the difference in magnitude. The equation that represents this problem is [latex]10^x=500,[\/latex]where <em>x<\/em> represents the difference in magnitudes on the <span id=\"term-00001\" class=\"no-emphasis\" data-type=\"term\">Richter Scale<\/span>. How would we solve for <em>x<\/em>?<\/p>\n<p id=\"fs-id1165135160312\">We have not yet learned a method for solving exponential equations. None of the algebraic tools discussed so far is sufficient to solve [latex]10^x=500.[\/latex] We know that [latex]10^2=100[\/latex] and [latex]10^3=1000,[\/latex] so it is clear that x must be some value between 2 and 3, since [latex]y=10^x[\/latex] is increasing. We can examine a graph, as in Figure 2, to better estimate the solution.<\/p>\n<p>&nbsp;<\/p>\n<figure id=\"attachment_955\" aria-describedby=\"caption-attachment-955\" style=\"width: 300px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-955\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.3-fig-2-300x294.webp\" alt=\"\" width=\"300\" height=\"294\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.3-fig-2-300x294.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.3-fig-2-65x64.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.3-fig-2-225x220.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.3-fig-2-350x343.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.3-fig-2.webp 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><figcaption id=\"caption-attachment-955\" class=\"wp-caption-text\">Figure 2<\/figcaption><\/figure>\n<p id=\"fs-id1165137662989\">Estimating from a graph, however, is imprecise. To find an algebraic solution, we must introduce a new function. Observe that the graph in Figure 2 passes the horizontal line test. The exponential function [latex]y=b^x[\/latex] is one-to-one, so its inverse, [latex]x=b^y[\/latex] is also a function. As is the case with all inverse functions, we simply interchange <em>x<\/em> and <em>y<\/em> and solve for <em>y<\/em> to find the inverse function. To represent <em>y<\/em> as a function of <em>x<\/em>, we use a logarithmic function of the form [latex]y=\\log_b(x)[\/latex] The base <em>b <\/em><strong>logarithm<\/strong> of a number is the exponent by which we must raise <em>b<\/em> to get that number.<\/p>\n<p id=\"fs-id1165137404844\">We read a logarithmic expression as, \u201cThe logarithm with base <em>b<\/em> of <em>x<\/em> is equal to <em>y<\/em>,&#8221; or, simplified, \u201clog base <em>b<\/em> of <em>x<\/em> is <em>y<\/em>.&#8221; We can also say, \u201c<em>b<\/em> raised to the power of <em>y<\/em> is <em>x<\/em>,&#8221; because logs are exponents. For example, the base 2 logarithm of 32 is 5, because 5 is the exponent we must apply to 2 to get 32. Since [latex]2^5=32,[\/latex] we can write [latex]\\log_232=5.[\/latex] We read this as \u201clog base 2 of 32 is 5.\u201d<\/p>\n<p id=\"fs-id1165137597501\">We can express the relationship between logarithmic form and its corresponding exponential form as follows:<\/p>\n<p style=\"text-align: center;\">[latex]\\log_b(x) = y \\Leftrightarrow b^y = x, \\quad b > 0, b \\neq 1[\/latex]<\/p>\n<p id=\"fs-id1165137678993\">Note that the base <em>b<\/em> is always positive.<\/p>\n<p><span id=\"fs-id1165137696233\" data-type=\"media\" data-alt=\"\" data-display=\"block\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-956 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.3-base-b-300x51.webp\" alt=\"\" width=\"300\" height=\"51\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.3-base-b-300x51.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.3-base-b-65x11.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.3-base-b-225x38.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.3-base-b-350x60.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.3-base-b.webp 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><br \/>\n<\/span><\/p>\n<p id=\"fs-id1165137400957\">Because logarithm is a function, it is most correctly written as [latex]\\log_b(x),[\/latex] using parentheses to denote function evaluation, just as we would with [latex]f(x).[\/latex] However, when the input is a single variable or number, it is common to see the parentheses dropped and the expression written without parentheses, as [latex]\\log_bx.[\/latex] Note that many calculators require parentheses around the <em>x<\/em>.<\/p>\n<p id=\"fs-id1165137827516\">We can illustrate the notation of logarithms as follows:<\/p>\n<p><span id=\"fs-id1165137771679\" data-type=\"media\" data-alt=\"\" data-display=\"block\"><br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"size-medium wp-image-957 aligncenter\" src=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.3-log-c-300x62.webp\" alt=\"\" width=\"300\" height=\"62\" srcset=\"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.3-log-c-300x62.webp 300w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.3-log-c-65x13.webp 65w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.3-log-c-225x47.webp 225w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.3-log-c-350x73.webp 350w, https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-content\/uploads\/sites\/234\/2025\/04\/6.3-log-c.webp 487w\" sizes=\"auto, (max-width: 300px) 100vw, 300px\" \/><br \/>\n<\/span><\/p>\n<p id=\"fs-id1165137575165\">Notice that, comparing the logarithm function and the exponential function, the input and the output are switched. This means [latex]y=\\log_b(x)[\/latex] and [latex]y=b^x[\/latex] are inverse functions.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Definition of the Logarithmic Function<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>A <strong>logarithm <\/strong>base <em>b<\/em> of a positive number <em>x <\/em>satisfies the following definition.<\/p>\n<p>For [latex]x> 0, b> 0, b\\not=1,[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]y=\\log_b(x)[\/latex] is\u00a0equivalent\u00a0to [latex]b_y=x[\/latex]<\/p>\n<p>where,<\/p>\n<ul>\n<li>we read [latex]\\log_b(x)[\/latex] as, \u201cthe logarithm with base <em>b <\/em>of <em>x<\/em>&#8221; or the \u201clog base <em>b<\/em> of <em>x<\/em>.&#8221;<\/li>\n<li>the logarithm <em>y<\/em> is the exponent to which <em>b<\/em> must be raised to get <em>x.<\/em><\/li>\n<\/ul>\n<p>Also, since the logarithmic and exponential functions switch the <em>x<\/em> and <em>y<\/em> values, the domain and range of the exponential function are interchanged for the logarithmic function. Therefore,<\/p>\n<ul>\n<li>the domain of the logarithm function with base <em>b<\/em> is [latex](0, \\infty).[\/latex]<\/li>\n<li>the range of the logarithm function with base <em>b<\/em> is [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\">Q&amp;A<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p><strong>Q: Can we take the logarithm of a negative number?<\/strong><\/p>\n<p><em data-effect=\"italics\">A: No. Because the base of an exponential function is always positive, no power of that base can ever be negative. We can never take the logarithm of a negative number. Also, we cannot take the logarithm of zero. Calculators may output a log of a negative number when in complex mode, but the log of a negative number is not a real number.<\/em><\/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 an equation in logarithmic form [latex]lob_b(x)=y,[\/latex] convert it to exponential form. <\/strong><\/p>\n<ol>\n<li>Examine the equation [latex]\\log_b(x),[\/latex] and identify b, y, and x.<\/li>\n<li>Rewrite [latex]\\log_b(x)=y[\/latex] as [latex]b^y=x.[\/latex]<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 1: Converting from Logarithmic Form to Exponential Form<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Write the following logarithmic equations in exponential form.<\/p>\n<p>(a) [latex]\\log_6(\\sqrt{6})=\\frac{1}{2}[\/latex]<\/p>\n<p>(b) [latex]\\log_3(9)=2[\/latex]<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>First, identify the values of <em>b<\/em>, <em>y<\/em>, and <em>x<\/em>. Then, write the equation in the form [latex]b^y=x.[\/latex]<\/p>\n<p>(a) [latex]\\log_6(\\sqrt{6})=\\frac{1}{2}[\/latex]<\/p>\n<p>Here, [latex]b=6, y=\\frac{1}{2},[\/latex] and [latex]x=\\sqrt{6}.[\/latex] Therefore, the equation [latex]\\log_6(\\sqrt{6})=\\frac{1}{2}[\/latex] is equivalent to [latex]6^{\\frac{1}{2}}=\\sqrt{6}.[\/latex]<\/p>\n<p>(b) [latex]\\log_3(9)=2[\/latex]<\/p>\n<p>Here, [latex]b=3, y=2,[\/latex] and [latex]x=9.[\/latex] Therefore, the equation [latex]\\log_3(9)=2[\/latex] is equivalent to [latex]3^2=9.[\/latex]<\/p>\n<\/details>\n<\/div>\n<\/div>\n<section><\/section>\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>Write the following logarithmic equations in exponential form.<\/p>\n<p>(a) [latex]\\log_{10}(1,000,000)=6[\/latex]<\/p>\n<p>(b) [latex]\\log_5(25)=2[\/latex]<\/p>\n<\/div>\n<\/div>\n<section id=\"fs-id1165137585244\" data-depth=\"1\">\n<h2 data-type=\"title\">Converting from Exponential to Logarithmic Form<\/h2>\n<p id=\"fs-id1165137933968\">To convert from exponents to logarithms, we follow the same steps in reverse. We identify the base <em>b<\/em>, exponent\u00a0<em>x<\/em>, and output <em>y<\/em>. Then we write [latex]x=\\log^b(y).[\/latex]<\/p>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 2: Converting from Exponential Form to Logarithmic Form<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Write the following exponential equations in logarithmic form.<\/p>\n<p>(a) [latex]2^3=8[\/latex]<\/p>\n<p>(b) [latex]5^2=25[\/latex]<\/p>\n<p>(c) [latex]10^{-4}=\\frac{1}{10,000}[\/latex]<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>First, identify the values of <em>b<\/em>, <em>y<\/em>, and <em>x<\/em>. Then, write the equation in the form [latex]x=\\log_b(y).[\/latex]<\/p>\n<p>(a) [latex]2^3=8[\/latex]<\/p>\n<p>Here, [latex]b=2, x=3,[\/latex] and [latex]y=8.[\/latex] Therefore, the equation [latex]2^3=8[\/latex] is equivalent to [latex]\\log_2(8)=3.[\/latex]<\/p>\n<p>(b) [latex]5^5=25[\/latex]<\/p>\n<p>Here, [latex]b=5, x=2,[\/latex] and [latex]y=25.[\/latex] Therefore, the equation [latex]5^2=25[\/latex] is equivalent to [latex]\\log_5(25)=2.[\/latex]<\/p>\n<p>(c) [latex]10^{-4}=\\frac{1}{10,000}[\/latex]<\/p>\n<p>Here, [latex]b=10, x=-4,[\/latex] and [latex]y=\\frac{1}{10,000}.[\/latex] Therefore, the equation [latex]10^{-4}=\\frac{1}{10,000}[\/latex] is equivalent to [latex]\\log_{10}\\left(\\frac{1}{10,000}\\right)=-4.[\/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 #2<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Write the following exponential equations in logarithmic form.<\/p>\n<p>(a) [latex]3^2=9[\/latex]<\/p>\n<p>(b) [latex]5^3=125[\/latex]<\/p>\n<p>(c) [latex]2^{-1}=\\frac{1}{2}[\/latex]<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165137530906\" data-depth=\"1\">\n<h2 data-type=\"title\">Evaluating Logarithms<\/h2>\n<p id=\"fs-id1165137422589\">Knowing the squares, cubes, and roots of numbers allows us to evaluate many logarithms mentally. For example, consider [latex]\\log_28.[\/latex] We ask, \u201cTo what exponent must 2 be raised in order to get 8?\u201d Because we already know [latex]2^3=8,[\/latex] it follows that [latex]log_28=3.[\/latex]<\/p>\n<p id=\"fs-id1165137733822\">Now consider solving [latex]\\log_749[\/latex] and [latex]\\log_327[\/latex] mentally.<\/p>\n<ul id=\"fs-id1165137937690\">\n<li>We ask, \u201cTo what exponent must 7 be raised in order to get 49?\u201d We know [latex]7^2=49.[\/latex] Therefore, [latex]\\log_749=2.[\/latex]<\/li>\n<li>We ask, \u201cTo what exponent must 3 be raised in order to get 27?\u201d We know [latex]3^3=27.[\/latex] Therefore, [latex]\\log_327=3.[\/latex]<\/li>\n<\/ul>\n<p id=\"fs-id1165137456358\">Even some seemingly more complicated logarithms can be evaluated without a calculator. For example, let\u2019s evaluate [latex]\\log_{\\frac{2}{3}}\\frac{4}{9}[\/latex] mentally.<\/p>\n<ul id=\"fs-id1165137584208\">\n<li>We ask, \u201cTo what exponent must [latex]\\frac{2}{3}[\/latex] be raised in order to get [latex]\\frac{4}{9}?\"[\/latex] We know [latex]2^2=4[\/latex] and [latex]3^2=9,[\/latex] so [latex]\\left(\\frac{2}{3}\\right)^2=\\frac{4}{9}.[\/latex] Therefore, [latex]\\log_{\\frac{2}{3}}\\left(\\frac{4}{9}\\right)=2.[\/latex]<\/li>\n<\/ul>\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 logarithm of the form [latex]y=\\log^b(x),[\/latex] evaluate it mentally.<\/strong><\/p>\n<ol>\n<li>Rewrite the argument <em>x<\/em> as a power of [latex]b: b^y=x.[\/latex]<\/li>\n<li>Use previous knowledge of powers of <em>b<\/em> identify <em>y<\/em> by asking, \u201cTo what exponent should b be raised in order to get x?&#8221;<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 3: Solving Logarithms Mentally<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Solve [latex]y=\\log_4(64)[\/latex] without using a calculator.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>First we rewrite the logarithm in exponential form: [latex]4^y=64.[\/latex] Next, we ask, \u201cTo what exponent must 4 be raised in order to get 64?\u201d<\/p>\n<p>We know<\/p>\n<p style=\"text-align: center;\">[latex]4^3=64[\/latex]<\/p>\n<p>Therefore,<\/p>\n<p style=\"text-align: center;\">[latex]\\log_4(64)=3[\/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>Solve [latex]y=\\log_{121}(11)[\/latex] without using a calculator.<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 4: Evaluating the Logarithm of a Reciprocal<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Evaluate [latex]y=\\log_3\\left(\\frac{1}{27}\\right)[\/latex] without using a calculator.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>First we rewrite the logarithm in exponential form: [latex]3^y=\\frac{1}{27}.[\/latex] Next, we ask, \u201cTo what exponent must 3 be raised in order to get [latex]\\frac{1}{27}?\"[\/latex]<\/p>\n<p>We know [latex]3^3=27,[\/latex] but what must we do to get the reciprocal, [latex]\\frac{1}{27}?[\/latex] Recall from working with exponents that [latex]b^{-a}=\\frac{1}{b^a}.[\/latex] We use this information to write<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{rl} 3^{-3} &=& \\frac{1}{3^3} \\\\ &=& \\frac{1}{27} \\end{array}[\/latex]<\/p>\n<p>Therefore, [latex]\\log_3\\left(\\frac{1}{27}\\right)=-3.[\/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 #4<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Evaluate [latex]y=\\log_2\\left(\\frac{1}{32}\\right)[\/latex] without using a calculator.<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165137547253\" data-depth=\"1\">\n<h2 data-type=\"title\">Using Common Logarithms<\/h2>\n<p id=\"fs-id1165137574205\">Sometimes you may see a logarithm written without a base. When you see one written this way, you need to look at the expression before evaluating it. It may be that the base you use doesn&#8217;t matter. If you find it in computer science, it often means [latex]\\log_2(x).[\/latex] However, in mathematics it almost always means the common logarithm of 10. In other words, the expression [latex]\\log(x)[\/latex] often means [latex]\\log_{10}(x).[\/latex]<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Definition of the Common Logarithm<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>A <strong>common logarithm<\/strong> is a logarithm with base 10. We can also write [latex]\\log_{10}(x)[\/latex] simply as [latex]\\log(x).[\/latex] The common logarithm of a positive number <em>x<\/em> satisfies the following definition.<\/p>\n<p>For [latex]x> 0,[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]y=\\log(x)[\/latex] is\u00a0equivalent\u00a0to [latex]y=\\log_{10}(x)[\/latex]<\/p>\n<p>We read [latex]log(x)[\/latex] as, \u201cthe logarithm with base 10 of <em>x<\/em>&#8221; or \u201clog base 10 of <em>x<\/em>.&#8221;<\/p>\n<p>The logarithm <em>y<\/em> is the exponent to which 10 must be raised to get <em>x<\/em>.<\/p>\n<\/div>\n<\/div>\n<p id=\"para-00002\">Currently, we use [latex]\\log_b(x), lg(x)[\/latex] as the common logarithm, [latex]\\operatorname{lb}(x)\u00a0[\/latex] as the binary logarithm, and [latex]\\ln(x)[\/latex] as the natural logarithm. Writing [latex]\\lg(x)[\/latex] without specifying a base is now considered bad form, despite being frequently found in older materials.<\/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 common logarithm of the form [latex]y=\\log(x),[\/latex] evaluate it mentally. <\/strong><\/p>\n<ol>\n<li>Rewrite the argument <em>x<\/em> as a power of [latex]10: 10^y=x.[\/latex]<\/li>\n<li>Use previous knowledge of powers of 10 to identify <em>y<\/em> by asking, \u201cTo what exponent must 10 be raised in order to get <em>x<\/em>?&#8221;<\/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: Finding the Value of a Common Logarithm Mentally<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Evaluate [latex]y=\\log(1000)[\/latex] without using a calculator.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>First we rewrite the logarithm in exponential form: [latex]10^y=1000.[\/latex] Next, we ask, \u201cTo what exponent must 10 be raised in order to get 1000?\u201d We know<\/p>\n<p style=\"text-align: center;\">[latex]10^3=1000[\/latex]<\/p>\n<p>Therefore, [latex]\\log(1000)=3.[\/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>Evaluate [latex]y=\\log(1,000,000)[\/latex]<\/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 common logarithm with the form [latex]y=\\log(x),[\/latex] evaluate it using a calculator. <\/strong><\/p>\n<ol>\n<li>Press <strong>[LOG]<\/strong>.<\/li>\n<li>Enter the value given for <em>x<\/em> followed by <strong>[ ) ]<\/strong>.<\/li>\n<li>Press <strong>[ENTER]<\/strong>.<\/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: Finding the Value of a Common Logarithm Using a Calculator<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Evaluate [latex]y=\\log(321)[\/latex] to four decimal places using a calculator.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<ul>\n<li>Press <strong>[LOG<\/strong><\/li>\n<li>Enter 321, followed by <strong>[ ) ]<\/strong>.<\/li>\n<li>Press <strong>[ENTER]<\/strong>.<\/li>\n<\/ul>\n<p>Rounding to four decimal places, [latex]\\log(321)\\approx 2.5065.[\/latex]<\/p>\n<h3>Analysis<\/h3>\n<p>Note that [latex]10^2=100[\/latex] and that [latex]10^3=1000.[\/latex] Since 321 is between 100 and 1000, we know that [latex]\\log(321)[\/latex] must be between [latex]\\log(100)[\/latex] and [latex]\\log(1000).[\/latex] This gives us the following:<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{ccc} 1000 &<& 321 &<& 1000 \\\\ 2 &<& 2.5065 &<& 3 \\end{array}[\/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 #6<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Evaluate [latex]y=\\log(123)[\/latex] to four decimal places using a calculator.<\/p>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 7: Rewriting and Solving a Real-World Exponential Model<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>The amount of energy released from one earthquake was 500 times greater than the amount of energy released from another. The equation [latex]10^x=500[\/latex] represents this situation, where x is the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes?<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<p>We begin by rewriting the exponential equation in logarithmic form.<\/p>\n<p style=\"text-align: center;\">[latex]\\begin{array}{lll} 10^x &=& 500 \\\\ \\log(500) &=& x && \\text{Use the definition of the common log.} \\end{array}[\/latex]<\/p>\n<p>Next we evaluate the logarithm using a calculator:<\/p>\n<ul>\n<li>Press <strong>[LOG]<\/strong>.<\/li>\n<li>Enter 500, followed by <strong>[ ) ]<\/strong>.<\/li>\n<li>Press <strong>[ENTER]<\/strong>.<\/li>\n<li>To the nearest thousandth, [latex]\\log(500)\\approx 2.699.[\/latex]<\/li>\n<\/ul>\n<p>The difference in magnitudes was about [latex]2.699.[\/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>The amount of energy released from one earthquake was 8,500 times greater than the amount of energy released from another. The equation [latex]10^x=8500[\/latex] represents this situation, where <em>x<\/em> is the difference in magnitudes on the Richter Scale. To the nearest thousandth, what was the difference in magnitudes?<\/p>\n<\/div>\n<\/div>\n<\/section>\n<section id=\"fs-id1165137405741\" data-depth=\"1\">\n<h2 data-type=\"title\">Using Natural Logarithms<\/h2>\n<p id=\"fs-id1165137661970\">The most frequently used base for logarithms is <em>e<\/em>, the value of which is approximately [latex]2.71828.[\/latex] Base <em>e<\/em> logarithms are important in calculus and some scientific applications; they are called <strong>natural logarithms<\/strong>. The base <em>e<\/em> logarithm, [latex]log_e(x),[\/latex] has its own notation, [latex]ln(x).[\/latex]<\/p>\n<p id=\"fs-id1165137473872\">Most values of [latex]ln(x)[\/latex] can be found only using a calculator. The major exception is that, because the logarithm of 1 is always 0 in any base, [latex]\\ln1=0.[\/latex] For other natural logarithms, we can use the [latex]\\ln[\/latex] key that can be found on most scientific calculators. We can also find the natural logarithm of any power of <em>e<\/em> using the inverse property of logarithms.<\/p>\n<div class=\"textbox textbox--examples\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Definition of the Natural Logarithm<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>A <strong>natural logarithm<\/strong> is a logarithm with base <em>e<\/em>. We write [latex]\\log_e(x)[\/latex] simply as [latex]\\ln(x).[\/latex] The natural logarithm of a positive number <em>x<\/em> satisfies the following definition.<\/p>\n<p>For [latex]x> 0,[\/latex]<\/p>\n<p style=\"text-align: center;\">[latex]y=\\ln(x)[\/latex] is\u00a0equivalent\u00a0to [latex]e^y=x.[\/latex]<\/p>\n<p>We read [latex]\\ln(x)[\/latex] as, \u201cthe logarithm with base <em>e<\/em> of <em>x<\/em>\u201d or \u201cthe natural logarithm of <em>x<\/em>.\u201d<\/p>\n<p>The logarithm <em>y<\/em> is the exponent to which <em>e<\/em> must be raised to get <em>x<\/em>.<\/p>\n<p>Since the functions [latex]y=e^x[\/latex] and [latex]y=\\ln(x)[\/latex] are inverse functions, [latex]\\ln(e^x)=x[\/latex] for all x and [latex]e^{\\ln(x)=x}=x[\/latex] for [latex]x> 0.[\/latex]<\/p>\n<\/div>\n<\/div>\n<div class=\"os-note-body\">\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 natural logarithm with the form [latex]y=\\ln(x),[\/latex] evaluate it using a calculator. <\/strong><\/p>\n<ol>\n<li>Press <strong>[LN]<\/strong>.<\/li>\n<li>Enter the value given for <em>x<\/em>, followed by <strong>[ ) ]<\/strong>.<\/li>\n<li>Press <strong>[ENTER]<\/strong>.<\/li>\n<\/ol>\n<\/div>\n<\/div>\n<div class=\"textbox textbox--exercises\">\n<header class=\"textbox__header\">\n<p class=\"textbox__title\">Example 8: Evaluating a Natural Logarithm Using a Calculator<\/p>\n<\/header>\n<div class=\"textbox__content\">\n<p>Evaluate [latex]y=\\ln(500)[\/latex] to four decimal places using a calculator.<\/p>\n<p>&nbsp;<\/p>\n<details>\n<summary><strong>Solution (click to expand)<\/strong><\/summary>\n<ul>\n<li>Press <strong>[LN]<\/strong>.<\/li>\n<li>Enter 500, followed by <strong>[ ) ]<\/strong>.<\/li>\n<li>Press <strong>[ENTER]<\/strong>.<\/li>\n<\/ul>\n<p>Rounding to four decimal places, [latex]\\ln(500)\\approx 6.2146.[\/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>Evaluate [latex]\\ln(-500).[\/latex]<\/p>\n<\/div>\n<\/div>\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 this online resource for additional instruction and practice with logarithms.<\/p>\n<ul>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=z296tOPj0HA\">Introduction to Logarithms<\/a><\/li>\n<\/ul>\n<\/div>\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\">6.3 Section Exercises<\/span><\/h2>\n<section id=\"fs-id1165135192789\" class=\"section-exercises\" data-depth=\"1\">\n<section id=\"fs-id1165137427076\" data-depth=\"2\">\n<h3 data-type=\"title\">Verbal<\/h3>\n<div id=\"fs-id1165137817361\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137559978\" 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-id1165137817361-solution\">1<\/a><span class=\"os-divider\">. <\/span>What is a base b logarithm? Discuss the meaning by interpreting each part of the equivalent equations [latex]b^y=x[\/latex] and [latex]\\log_bx=y[\/latex] for [latex]b> 0, b\\not=1.[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137574896\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137658231\" data-type=\"problem\">\n<p><span class=\"os-number\">2<\/span><span class=\"os-divider\">. <\/span>How is the logarithmic function [latex]f(x)=\\log_bx[\/latex] related to the exponential function [latex]g(x)=b^x?[\/latex] What is the result of composing these two functions?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137446568\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137602953\" 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-id1165137446568-solution\">3<\/a><span class=\"os-divider\">. <\/span>How can the logarithmic equation [latex]\\log_bx=y[\/latex] be solved for x using the properties of exponents?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137470358\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135526986\" data-type=\"problem\">\n<p><span class=\"os-number\">4<\/span><span class=\"os-divider\">.\u00a0<\/span>Discuss the meaning of the common logarithm. What is its relationship to a logarithm with base <em>b <\/em>and how does the notation differ?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137507578\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137828407\" 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-id1165137507578-solution\">5<\/a><span class=\"os-divider\">. <\/span>Discuss the meaning of the natural logarithm. What is its relationship to a logarithm with base <em>b<\/em> and how does the notation differ?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<section id=\"fs-id1165137447239\" data-depth=\"2\">\n<h3 data-type=\"title\">Algebraic<\/h3>\n<p id=\"fs-id1165137414571\">For the following exercises, rewrite each equation in exponential form.<\/p>\n<div id=\"fs-id1165137646887\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137664870\" data-type=\"problem\">\n<p><span class=\"os-number\">6<\/span><span class=\"os-divider\">. <\/span> [latex]\\log_4(q)=m[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<div id=\"fs-id1165137473551\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137454663\" 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-id1165137473551-solution\">7<\/a><span class=\"os-divider\">. <\/span> [latex]\\log_a(b)=c[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137506749\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135536326\" data-type=\"problem\">\n<p><span class=\"os-number\">8<\/span><span class=\"os-divider\">. <\/span> [latex]\\log_{16}(y)=x[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137410961\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137602136\" 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-id1165137410961-solution\">9<\/a><span class=\"os-divider\">. <\/span> [latex]\\log_x(64)=y[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137673422\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135585639\" data-type=\"problem\">\n<p><span class=\"os-number\">10<\/span><span class=\"os-divider\">. <\/span> [latex]\\log_y(x)=-11[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135356538\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137501385\" 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-id1165135356538-solution\">11<\/a><span class=\"os-divider\">. <\/span> [latex]\\log_{15}(a)=b[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137580855\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137438413\" data-type=\"problem\">\n<p><span class=\"os-number\">12<\/span><span class=\"os-divider\">. <\/span> [latex]\\log_y(137)=x[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137611652\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134038222\" 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-id1165137611652-solution\">13<\/a><span class=\"os-divider\">. <\/span> [latex]\\log_{13}(142)=a[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137724172\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137724174\" data-type=\"problem\">\n<p><span class=\"os-number\">14<\/span><span class=\"os-divider\">. <\/span> [latex]\\log(v)=t[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137735890\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137735892\" 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-id1165137735890-solution\">15<\/a><span class=\"os-divider\">. <\/span> [latex]\\ln(w)=n[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165137810335\">For the following exercises, rewrite each equation in logarithmic form.<\/p>\n<div id=\"fs-id1165135186264\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135203858\" data-type=\"problem\">\n<p><span class=\"os-number\">16<\/span><span class=\"os-divider\">. <\/span> [latex]4^x=y[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135194695\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137724816\" 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-id1165135194695-solution\">17<\/a><span class=\"os-divider\">. <\/span> [latex]c^d=k[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137698394\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137434093\" data-type=\"problem\">\n<p><span class=\"os-number\">18<\/span><span class=\"os-divider\">. <\/span> [latex]m^{-7}=n[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135384408\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135384410\" 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-id1165135384408-solution\">19<\/a><span class=\"os-divider\">. <\/span> [latex]19^x=y[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135358961\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137627327\" data-type=\"problem\">\n<p><span class=\"os-number\">20<\/span><span class=\"os-divider\">. <\/span> [latex]x^{=\\frac{10}{13}}=y[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165132971714\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165132971716\" 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-id1165132971714-solution\">21<\/a><span class=\"os-divider\">. <\/span> [latex]n^4=103[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137564489\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137564491\" data-type=\"problem\">\n<p><span class=\"os-number\">22<\/span><span class=\"os-divider\">. <\/span> [latex]\\left(\\frac{7}{5}\\right)^m=n[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137731983\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137597971\" 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-id1165137731983-solution\">23<\/a><span class=\"os-divider\">. <\/span> [latex]y^x=\\frac{39}{100}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137827739\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137762434\" data-type=\"problem\">\n<p><span class=\"os-number\">24<\/span><span class=\"os-divider\">. <\/span> [latex]10^a=b[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135296148\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165134040600\" 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-id1165135296148-solution\">25<\/a><span class=\"os-divider\">. <\/span> [latex]e^k=h[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165137762947\">For the following exercises, solve for <em>x<\/em> by converting the logarithmic equation to exponential form.<\/p>\n<div id=\"fs-id1165137438777\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137438779\" data-type=\"problem\">\n<p><span class=\"os-number\">26<\/span><span class=\"os-divider\">. <\/span> [latex]\\log_3(x)=2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137482919\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137628211\" 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-id1165137482919-solution\">27<\/a><span class=\"os-divider\">. <\/span> [latex]\\log_2(x)=-3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137726345\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137726348\" data-type=\"problem\">\n<p><span class=\"os-number\">28<\/span><span class=\"os-divider\">. <\/span> [latex]\\log_5(x)=2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137389073\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137639044\" 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-id1165137389073-solution\">29<\/a><span class=\"os-divider\">. <\/span> [latex]\\log_3(x)=3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137661950\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137726952\" data-type=\"problem\">\n<p><span class=\"os-number\">30<\/span><span class=\"os-divider\">. <\/span> [latex]\\log_2(x)=6[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137627690\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137657454\" 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-id1165137627690-solution\">31<\/a><span class=\"os-divider\">. <\/span> [latex]\\log_9(x)=\\frac{1}{2}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135203739\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135203742\" data-type=\"problem\">\n<p><span class=\"os-number\">32<\/span><span class=\"os-divider\">. <\/span> [latex]\\log_{18}(x)=2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135613168\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137414109\" 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-id1165135613168-solution\">33<\/a><span class=\"os-divider\">. <\/span> [latex]\\log_6(x)=-3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137937666\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137937668\" data-type=\"problem\">\n<p><span class=\"os-number\">34<\/span><span class=\"os-divider\">. <\/span> [latex]\\log(x)=3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137736080\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137442469\" 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-id1165137736080-solution\">35<\/a><span class=\"os-divider\">. <\/span> [latex]\\ln(x)=2[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165137634430\">For the following exercises, use the definition of common and natural logarithms to simplify.<\/p>\n<div id=\"fs-id1165137634434\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137732545\" data-type=\"problem\">\n<p><span class=\"os-number\">36<\/span><span class=\"os-divider\">. <\/span> [latex]\\log(100^8)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137936925\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137936927\" 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-id1165137936925-solution\">37<\/a><span class=\"os-divider\">. <\/span> [latex]10^{\\log(32)}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137529000\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137529002\" data-type=\"problem\">\n<p><span class=\"os-number\">38<\/span><span class=\"os-divider\">. <\/span> [latex]32\\log(.0001)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165134079645\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137566210\" 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-id1165134079645-solution\">39<\/a><span class=\"os-divider\">. <\/span> [latex]e^{\\ln(1.06)}[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137400162\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137400164\" data-type=\"problem\">\n<p><span class=\"os-number\">40<\/span><span class=\"os-divider\">. <\/span> [latex]\\ln(e^{-5.03})[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137543862\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137543865\" 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-id1165137543862-solution\">41<\/a><span class=\"os-divider\">. <\/span> [latex]e^{\\ln(10.125)}+4[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<section id=\"fs-id1165137451809\" data-depth=\"2\">\n<h3 data-type=\"title\">Numeric<\/h3>\n<p id=\"fs-id1165137564770\">For the following exercises, evaluate the base <em>b<\/em> logarithmic expression without using a calculator.<\/p>\n<div id=\"fs-id1165137676437\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137676439\" data-type=\"problem\">\n<p><span class=\"os-number\">42<\/span><span class=\"os-divider\">. <\/span> [latex]\\log_3\\left(\\frac{1}{27}\\right)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<div id=\"fs-id1165137590077\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137590079\" 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-id1165137590077-solution\">43<\/a><span class=\"os-divider\">. <\/span> [latex]\\log_6(\\sqrt{6})[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137415151\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137423807\" data-type=\"problem\">\n<p><span class=\"os-number\">44<\/span><span class=\"os-divider\">. <\/span> [latex]\\log_2\\left(\\frac{1}{8}\\right)+4[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135190140\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137806053\" 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-id1165135190140-solution\">45<\/a><span class=\"os-divider\">. <\/span> [latex]6\\log_8(4)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165137532658\">For the following exercises, evaluate the common logarithmic expression without using a calculator.<\/p>\n<div id=\"fs-id1165137542512\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135613435\" data-type=\"problem\">\n<p><span class=\"os-number\">46<\/span><span class=\"os-divider\">. <\/span> [latex]\\log(10,000)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137755960\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137736340\" 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-id1165137755960-solution\">47<\/a><span class=\"os-divider\">. <\/span> [latex]\\log(0.001)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135190197\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135190199\" data-type=\"problem\">\n<p><span class=\"os-number\">48<\/span><span class=\"os-divider\">. <\/span> [latex]\\log(1)+7[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137432736\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137655695\" 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-id1165137432736-solution\">49<\/a><span class=\"os-divider\">. <\/span> [latex]2\\log(100^{-3})[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<p id=\"fs-id1165137570442\">For the following exercises, evaluate the natural logarithmic expression without using a calculator.<\/p>\n<div id=\"fs-id1165137619029\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137619032\" data-type=\"problem\">\n<p><span class=\"os-number\">50<\/span><span class=\"os-divider\">. <\/span> [latex]\\ln(e^{\\frac{1}{3}})[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137736480\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137736482\" 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-id1165137736480-solution\">51<\/a><span class=\"os-divider\">. <\/span> [latex]\\ln(1)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137804477\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137474568\" data-type=\"problem\">\n<p><span class=\"os-number\">52<\/span><span class=\"os-divider\">. <\/span> [latex]\\ln(e^{-0.225})-3[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137762957\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135570229\" 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-id1165137762957-solution\">53<\/a><span class=\"os-divider\">. <\/span> [latex]25\\ln(e^{\\frac{2}{5}})[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<section id=\"fs-id1165137527430\" data-depth=\"2\">\n<h3 data-type=\"title\">Technology<\/h3>\n<p id=\"fs-id1165137401108\">For the following exercises, evaluate each expression using a calculator. Round to the nearest thousandth.<\/p>\n<div id=\"fs-id1165137626617\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137626619\" data-type=\"problem\">\n<p><span class=\"os-number\">54<\/span><span class=\"os-divider\">. <\/span> [latex]\\log(0.04)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<div id=\"fs-id1165137640432\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137640434\" 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-id1165137640432-solution\">55<\/a><span class=\"os-divider\">. <\/span> [latex]\\ln(15)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137731720\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137731722\" data-type=\"problem\">\n<p><span class=\"os-number\">56<\/span><span class=\"os-divider\">. <\/span> [latex]\\ln\\left(\\frac{4}{5}\\right)[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137527828\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137527830\" 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-id1165137527828-solution\">57<\/a><span class=\"os-divider\">. <\/span> [latex]\\log(\\sqrt{2})[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137655087\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137655089\" data-type=\"problem\">\n<p><span class=\"os-number\">58<\/span><span class=\"os-divider\">. <\/span> [latex]\\ln(\\sqrt{2})[\/latex]<\/p>\n<\/div>\n<\/section>\n<\/div>\n<section id=\"fs-id1165137932472\" data-depth=\"2\">\n<h3 data-type=\"title\">Extensions<\/h3>\n<div id=\"fs-id1165137435099\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137593336\" 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-id1165137435099-solution\">59<\/a><span class=\"os-divider\">. <\/span>Is [latex]x=0[\/latex] in the domain of the function [latex]f(x)=\\log(x)?[\/latex] If so, what is the value of the function when [latex]x=0?[\/latex] Verify the result.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<div id=\"fs-id1165137597455\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137597458\" data-type=\"problem\">\n<p><span class=\"os-number\">60<\/span><span class=\"os-divider\">. <\/span>Is [latex]f(x)=0[\/latex] in the range of the function [latex]f(x)=\\log(x)?[\/latex] If so, for what value of <em>x<\/em>? Verify the result.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137548869\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137548871\" 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-id1165137548869-solution\">61<\/a><span class=\"os-divider\">. <\/span>Is there a number <em>x<\/em> such that [latex]\\lnx=2?[\/latex] If so, what is that number? Verify the result.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137640809\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137603717\" data-type=\"problem\">\n<p><span class=\"os-number\">62<\/span><span class=\"os-divider\">. <\/span>Is the following true: [latex]\\frac{\\log_3(27)}{log_4\\left(\\frac{1}{64}\\right)}=-1?[\/latex] Verify the result.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165137453751\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137453754\" 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-id1165137453751-solution\">63<\/a><span class=\"os-divider\">. <\/span>Is the following true: [latex]\\frac{\\ln(e^{1.725})}{ln(1)}=1.725?[\/latex] Verify the result.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<section id=\"fs-id1165135437184\" data-depth=\"2\">\n<h3 data-type=\"title\">Real-World Applications<\/h3>\n<div id=\"fs-id1165137732791\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137913993\" data-type=\"problem\">\n<p><span class=\"os-number\">64<\/span><span class=\"os-divider\">. <\/span>The exposure index [latex]EI[\/latex] for a camera is a measurement of the amount of light that hits the image receptor. It is determined by the equation [latex]EI=\\log_2\\left(\\frac{f^2}{t}\\right),[\/latex] where f is the \u201cf-stop\u201d setting on the camera, and t is the exposure time in seconds. Suppose the f-stop setting is 8 and the desired exposure time is 2 seconds. What will the resulting exposure index be?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<div id=\"fs-id1165137602359\" class=\"os-hasSolution\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165137838750\" 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-id1165137602359-solution\">65<\/a><span class=\"os-divider\">. <\/span>Refer to the previous exercise. Suppose the light meter on a camera indicates an [latex]EI[\/latex] of -2, and the desired exposure time is 16 seconds. What should the f-stop setting be?<\/p>\n<\/div>\n<\/section>\n<\/div>\n<div id=\"fs-id1165135192763\" data-type=\"exercise\">\n<header><\/header>\n<section>\n<div id=\"fs-id1165135192765\" data-type=\"problem\">\n<p><span class=\"os-number\">66<\/span><span class=\"os-divider\">. <\/span>The intensity levels <em data-effect=\"italics\">I<\/em> of two earthquakes measured on a seismograph can be compared by the formula [latex]\\log\\frac{I_1}{I_2}=M_1-M_2[\/latex] where [latex]M[\/latex] is the magnitude given by the Richter Scale. In August 2009, an earthquake of magnitude 6.1 hit Honshu, Japan. In March 2011, that same region experienced yet another, more devastating earthquake, this time with a magnitude of 9.0.<sup id=\"footnote-ref5\" data-type=\"footnote-number\"><\/sup> How many times greater was the intensity of the 2011 earthquake? Round to the nearest whole number.<\/p>\n<\/div>\n<\/section>\n<\/div>\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-240","chapter","type-chapter","status-publish","hentry"],"part":160,"_links":{"self":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/240","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":11,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/240\/revisions"}],"predecessor-version":[{"id":1652,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapters\/240\/revisions\/1652"}],"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\/240\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/media?parent=240"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/pressbooks\/v2\/chapter-type?post=240"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/contributor?post=240"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.ccconline.org\/ccacollegealgebra\/wp-json\/wp\/v2\/license?post=240"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}