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41 Graphs of Logarithmic Functions

Learning Objectives

In this section, you will:

  • Identify the domain of a logarithmic function.
  • Graph logarithmic functions.

In Graphs of Exponential Functions, we saw how creating a graphical representation of an exponential model gives us another layer of insight for predicting future events. How do logarithmic graphs give us insight into situations? Because every logarithmic function is the inverse function of an exponential function, we can think of every output on a logarithmic graph as the input for the corresponding inverse exponential equation. In other words, logarithms give the cause for an effect.

To illustrate, suppose we invest$2500in an account that offers an annual interest rate of5compounded continuously. We already know that the balance in our account for any yeartcan be found with the equationA=2500e0.05t.

But what if we wanted to know the year for any balance? We would need to create a corresponding new function by interchanging the input and the output; thus we would need to create a logarithmic model for this situation. By graphing the model, we can see the output (year) for any input (account balance). For instance, what if we wanted to know how many years it would take for our initial investment to double? (Figure) shows this point on the logarithmic graph.

A graph titled, “Logarithmic Model Showing Years as a Function of the Balance in the Account”. The x-axis is labeled, “Account Balance”, and the y-axis is labeled, “Years”. The line starts at 💲25,000 on the first year. The graph also notes that the balance reaches 💲5,000 near year 14.
Figure 1.

In this section we will discuss the values for which a logarithmic function is defined, and then turn our attention to graphing the family of logarithmic functions.

Finding the Domain of a Logarithmic Function

Before working with graphs, we will take a look at the domain (the set of input values) for which the logarithmic function is defined.

Recall that the exponential function is defined asy=bxfor any real numberxand constantb>0, b1, where

  • The domain ofyis(,).
  • The range ofyis(0,).

In the last section we learned that the logarithmic functiony=logb(x)is the inverse of the exponential functiony=bx.So, as inverse functions:

  • The domain ofy=logb(x)is the range ofy=bx:(0,).
  • The range ofy=logb(x)is the domain ofy=bx:(,).

Transformations of the parent functiony=logb(x)behave similarly to those of other functions. Just as with other parent functions, we can apply the four types of transformations—shifts, stretches, compressions, and reflections—to the parent function without loss of shape.

In Graphs of Exponential Functions we saw that certain transformations can change the range ofy=bx.Similarly, applying transformations to the parent functiony=logb(x)can change the domain. When finding the domain of a logarithmic function, therefore, it is important to remember that the domain consists only of positive real numbers. That is, the argument of the logarithmic function must be greater than zero.

For example, considerf(x)=log4(2x3).This function is defined for any values ofxsuch that the argument, in this case2x3, is greater than zero. To find the domain, we set up an inequality and solve forx:

2x3>0Show the argument greater than zero.2x>3Add 3.x>1.5Divide by 2.

In interval notation, the domain off(x)=log4(2x3)is(1.5,).

How To

Given a logarithmic function, identify the domain.

  1. Set up an inequality showing the argument greater than zero.
  2. Solve forx.
  3. Write the domain in interval notation.

Identifying the Domain of a Logarithmic Shift

What is the domain off(x)=log2(x+3)?

[reveal-answer q=”fs-id1165137894538″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165137894538″]

The logarithmic function is defined only when the input is positive, so this function is defined whenx+3>0.Solving this inequality,

x+3>0The input must be positive.x>3Subtract 3.

The domain off(x)=log2(x+3)is(3,).[/hidden-answer]

Try It

What is the domain off(x)=log5(x2)+1?

[reveal-answer q=”fs-id1165137651038″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165137651038″]

(2,)

[/hidden-answer]

Identifying the Domain of a Logarithmic Shift and Reflection

What is the domain off(x)=log(52x)?

[reveal-answer q=”fs-id1165137925255″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165137925255″]

The logarithmic function is defined only when the input is positive, so this function is defined when52x>0.Solving this inequality,

52x>0The input must be positive.2x>5Subtract 5.x<52Divide by 2 and switch the inequality.

The domain off(x)=log(52x)is(,52).[/hidden-answer]

Try It

What is the domain off(x)=log(x5)+2?

[reveal-answer q=”fs-id1165135187083″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165135187083″]

(5,)

[/hidden-answer]

Graphing Logarithmic Functions

Now that we have a feel for the set of values for which a logarithmic function is defined, we move on to graphing logarithmic functions. The family of logarithmic functions includes the parent functiony=logb(x)along with all its transformations: shifts, stretches, compressions, and reflections.

We begin with the parent functiony=logb(x).Because every logarithmic function of this form is the inverse of an exponential function with the formy=bx, their graphs will be reflections of each other across the liney=x.To illustrate this, we can observe the relationship between the input and output values ofy=2xand its equivalentx=log2(y)in (Figure).

x 3 2 1 0 1 2 3
2x=y 18 14 12 1 2 4 8
log2(y)=x 3 2 1 0 1 2 3

Using the inputs and outputs from (Figure), we can build another table to observe the relationship between points on the graphs of the inverse functionsf(x)=2xandg(x)=log2(x).See (Figure).

f(x)=2x (3,18) (2,14) (1,12) (0,1) (1,2) (2,4) (3,8)
g(x)=log2(x) (18,3) (14,2) (12,1) (1,0) (2,1) (4,2) (8,3)

As we’d expect, the x– and y-coordinates are reversed for the inverse functions. (Figure) shows the graph offandg.

Graph of two functions, f(x)=2^x and g(x)=log_2(x), with the line y=x denoting the axis of symmetry.
Figure 2. Notice that the graphs off(x)=2xandg(x)=log2(x)are reflections about the liney=x.

Observe the following from the graph:

  • f(x)=2xhas a y-intercept at(0,1)andg(x)=log2(x)has an x– intercept at(1,0).
  • The domain off(x)=2x, (,), is the same as the range ofg(x)=log2(x).
  • The range off(x)=2x, (0,), is the same as the domain ofg(x)=log2(x).

Characteristics of the Graph of the Parent Function, f(x) = logb(x)

For any real numberxand constantb>0,b1, we can see the following characteristics in the graph off(x)=logb(x):

  • one-to-one function
  • vertical asymptote:x=0
  • domain:(0,)
  • range:(,)
  • x-intercept:(1,0)and key point (b,1)
  • y-intercept: none
  • increasing ifb>1
  • decreasing if[latex]\,0

See (Figure).

Figure 3.

(Figure) shows how changing the basebinf(x)=logb(x)can affect the graphs. Observe that the graphs compress vertically as the value of the base increases. (Note: recall that the functionln(x)has basee2.718.)

Graph of three equations: y=log_2(x) in blue, y=ln(x) in orange, and y=log(x) in red. The y-axis is the asymptote.
Figure 4. The graphs of three logarithmic functions with different bases, all greater than 1.

How To

Given a logarithmic function with the formf(x)=logb(x), graph the function.

  1. Draw and label the vertical asymptote,x=0.
  2. Plot the x-intercept,(1,0).
  3. Plot the key point(b,1).
  4. Draw a smooth curve through the points.
  5. State the domain,(0,),the range,(,),and the vertical asymptote,x=0.

Graphing a Logarithmic Function with the Form f(x) = logb(x).

Graphf(x)=log5(x).State the domain, range, and asymptote.

[reveal-answer q=”fs-id1165137501967″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165137501967″]

Before graphing, identify the behavior and key points for the graph.

  • Sinceb=5is greater than one, we know the function is increasing. The left tail of the graph will approach the vertical asymptotex=0, and the right tail will increase slowly without bound.
  • The x-intercept is(1,0).
  • The key point(5,1)is on the graph.
  • We draw and label the asymptote, plot and label the points, and draw a smooth curve through the points (see (Figure)).
Graph of f(x)=log_5(x) with labeled points at (1, 0) and (5, 1). The y-axis is the asymptote.
Figure 5.

The domain is(0,), the range is(,), and the vertical asymptote isx=0.[/hidden-answer]

Try It

Graphf(x)=log15(x).State the domain, range, and asymptote.

[reveal-answer q=”fs-id1165137407451″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165137407451″]Graph of f(x)=log_(1/5)(x) with labeled points at (1/5, 1) and (1, 0). The y-axis is the asymptote.

The domain is(0,),the range is(,), and the vertical asymptote isx=0.

[/hidden-answer]

Graphing Transformations of Logarithmic Functions

As we mentioned in the beginning of the section, transformations of logarithmic graphs behave similarly to those of other parent functions. We can shift, stretch, compress, and reflect the parent functiony=logb(x)without loss of shape.

Graphing a Horizontal Shift of f(x) = logb(x)

When a constantcis added to the input of the parent functionf(x)=logb(x), the result is a horizontal shiftcunits in the opposite direction of the sign onc.To visualize horizontal shifts, we can observe the general graph of the parent functionf(x)=logb(x)and forc>0alongside the shift left,g(x)=logb(x+c), and the shift right,h(x)=logb(xc). See (Figure).

Graph of two functions. The parent function is f(x)=log_b(x), with an asymptote at x=0 and g(x)=log_b(x+c) is the translation function with an asymptote at x=-c. This shows the translation of shifting left.
Figure 6.

Horizontal Shifts of the Parent Function y = logb(x)

For any constantc,the functionf(x)=logb(x+c)

  • shifts the parent functiony=logb(x)leftcunits ifc>0.
  • shifts the parent functiony=logb(x)rightcunits ifc<0.
  • has the vertical asymptotex=c.
  • has domain(c,).
  • has range(,).

How To

Given a logarithmic function with the formf(x)=logb(x+c), graph the translation.

  1. Identify the horizontal shift:
    1. Ifc>0,shift the graph off(x)=logb(x)leftcunits.
    2. Ifc<0,shift the graph off(x)=logb(x)rightcunits.
  2. Draw the vertical asymptotex=c.
  3. Identify three key points from the parent function. Find new coordinates for the shifted functions by subtractingcfrom thexcoordinate.
  4. Label the three points.
  5. The Domain is(c,),the range is(,), and the vertical asymptote isx=c.

Graphing a Horizontal Shift of the Parent Function y = logb(x)

Sketch the horizontal shiftf(x)=log3(x2)alongside its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.

[reveal-answer q=”fs-id1165137759883″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165137759883″]

Since the function isf(x)=log3(x2), we noticex+(2)=x2.

Thusc=2,soc<0.This means we will shift the functionf(x)=log3(x)right 2 units.

The vertical asymptote isx=(2)orx=2.

Consider the three key points from the parent function,(13,1),(1,0),and(3,1).

The new coordinates are found by adding 2 to thexcoordinates.

Label the points(73,1),(3,0),and(5,1).

The domain is(2,),the range is(,),and the vertical asymptote isx=2.

Graph of two functions. The parent function is y=log_3(x), with an asymptote at x=0 and labeled points at (1/3, -1), (1, 0), and (3, 1).The translation function f(x)=log_3(x-2) has an asymptote at x=2 and labeled points at (3, 0) and (5, 1).
Figure 7.

[/hidden-answer]

Try It

Sketch a graph off(x)=log3(x+4)alongside its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.

[reveal-answer q=”fs-id1165134373485″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165134373485″]Graph of two functions. The parent function is y=log_3(x), with an asymptote at x=0 and labeled points at (1, 0), and (3, 1).The translation function f(x)=log_3(x+4) has an asymptote at x=-4 and labeled points at (-3, 0) and (-1, 1).

The domain is(4,),the range(,),and the asymptotex=4.

[/hidden-answer]

Graphing a Vertical Shift of y = logb(x)

When a constantdis added to the parent functionf(x)=logb(x),the result is a vertical shiftdunits in the direction of the sign ond.To visualize vertical shifts, we can observe the general graph of the parent functionf(x)=logb(x)alongside the shift up,g(x)=logb(x)+dand the shift down,h(x)=logb(x)d.See (Figure).

Graph of two functions. The parent function is f(x)=log_b(x), with an asymptote at x=0 and g(x)=log_b(x)+d is the translation function with an asymptote at x=0. This shows the translation of shifting up. Graph of two functions. The parent function is f(x)=log_b(x), with an asymptote at x=0 and g(x)=log_b(x)-d is the translation function with an asymptote at x=0. This shows the translation of shifting down.
Figure 8.

Vertical Shifts of the Parent Function y = logb(x)

For any constantd,the functionf(x)=logb(x)+d

  • shifts the parent functiony=logb(x)updunits ifd>0.
  • shifts the parent functiony=logb(x)downdunits ifd<0.
  • has the vertical asymptotex=0.
  • has domain(0,).
  • has range(,).

How To

Given a logarithmic function with the formf(x)=logb(x)+d, graph the translation.

  1. Identify the vertical shift:
    • Ifd>0, shift the graph off(x)=logb(x)upd units.
    • Ifd<0, shift the graph off(x)=logb(x)downd units.
  2. Draw the vertical asymptotex=0.
  3. Identify three key points from the parent function. Find new coordinates for the shifted functions by addingdto theycoordinate.
  4. Label the three points.
  5. The domain is(0,),the range is(,),and the vertical asymptote isx=0.

Graphing a Vertical Shift of the Parent Function y = logb(x)

Sketch a graph off(x)=log3(x)2alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.

[reveal-answer q=”fs-id1165137925369″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165137925369″]

Since the function isf(x)=log3(x)2,we will noticed=2.Thusd<0.

This means we will shift the functionf(x)=log3(x)down 2 units.

The vertical asymptote isx=0.

Consider the three key points from the parent function,(13,1),(1,0),and(3,1).

The new coordinates are found by subtracting 2 from the y coordinates.

Label the points(13,3),(1,2), and(3,1).

The domain is(0,),the range is(,), and the vertical asymptote isx=0.

Graph of two functions. The parent function is y=log_3(x), with an asymptote at x=0 and labeled points at (1/3, -1), (1, 0), and (3, 1).The translation function f(x)=log_3(x)-2 has an asymptote at x=0 and labeled points at (1, 0) and (3, 1).
Figure 9.

The domain is(0,),the range is(,),and the vertical asymptote isx=0.[/hidden-answer]

Try It

Sketch a graph off(x)=log2(x)+2alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.

[reveal-answer q=”676362″]Show Solution[/reveal-answer]
[hidden-answer a=”676362″]Graph of two functions. The parent function is y=log_2(x), with an asymptote at x=0 and labeled points at (1, 0), and (2, 1).The translation function f(x)=log_2(x)+2 has an asymptote at x=0 and labeled points at (0.25, 0) and (0.5, 1).

The domain is(0,),the range is(,),and the vertical asymptote isx=0.

[/hidden-answer]

Graphing Stretches and Compressions of y = logb(x)

When the parent functionf(x)=logb(x)is multiplied by a constanta>0, the result is a vertical stretch or compression of the original graph. To visualize stretches and compressions, we seta>1and observe the general graph of the parent functionf(x)=logb(x)alongside the vertical stretch,g(x)=alogb(x)and the vertical compression,h(x)=1alogb(x).See (Figure).

Figure 10.

Vertical Stretches and Compressions of the Parent Function y = logb(x)

For any constanta>1,the functionf(x)=alogb(x)

  • stretches the parent functiony=logb(x)vertically by a factor ofaifa>1.
  • compresses the parent functiony=logb(x)vertically by a factor ofaif[latex]\,0
  • has the vertical asymptotex=0.
  • has the x-intercept(1,0).
  • has domain(0,).
  • has range(,).

Given a logarithmic function with the formf(x)=alogb(x),a>0,graph the translation.

  1. Identify the vertical stretch or compressions:
    • If|a|>1,the graph off(x)=logb(x)is stretched by a factor ofaunits.
    • If|a|<1,the graph off(x)=logb(x)is compressed by a factor ofaunits.
  2. Draw the vertical asymptotex=0.
  3. Identify three key points from the parent function. Find new coordinates for the shifted functions by multiplying theycoordinates bya.
  4. Label the three points.
  5. The domain is(0,),the range is(,),and the vertical asymptote isx=0.

Graphing a Stretch or Compression of the Parent Function y = logb(x)

Sketch a graph off(x)=2log4(x)alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.

[reveal-answer q=”420184″]Show Solution[/reveal-answer]
[hidden-answer a=”420184″]Since the function isf(x)=2log4(x),we will noticea=2.

This means we will stretch the functionf(x)=log4(x)by a factor of 2.

The vertical asymptote isx=0.

Consider the three key points from the parent function,(14,1),(1,0),and(4,1).

The new coordinates are found by multiplying theycoordinates by 2.

Label the points(14,2),(1,0), and(4,2).

The domain is(0,), the range is(,),and the vertical asymptote isx=0.See (Figure).

Graph of two functions. The parent function is y=log_4(x), with an asymptote at x=0 and labeled points at (1, 0), and (4, 1).The translation function f(x)=2log_4(x) has an asymptote at x=0 and labeled points at (1, 0) and (2, 1).
Figure 11.

The domain is(0,), the range is(,), and the vertical asymptote isx=0.

[/hidden-answer]

Try It

Sketch a graph off(x)=12log4(x)alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.

[reveal-answer q=”fs-id1165137778952″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165137778952″]Graph of two functions. The parent function is y=log_4(x), with an asymptote at x=0 and labeled points at (1, 0), and (4, 1).The translation function f(x)=(1/2)log_4(x) has an asymptote at x=0 and labeled points at (1, 0) and (16, 1).

The domain is(0,),the range is(,),and the vertical asymptote isx=0.

[/hidden-answer]

Combining a Shift and a Stretch

Sketch a graph off(x)=5log(x+2).State the domain, range, and asymptote.

[reveal-answer q=”fs-id1165137935559″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165137935559″]

Remember: what happens inside parentheses happens first. First, we move the graph left 2 units, then stretch the function vertically by a factor of 5, as in (Figure). The vertical asymptote will be shifted tox=2.The x-intercept will be(1,0).The domain will be(2,).Two points will help give the shape of the graph:(1,0)and(8,5).We chosex=8as the x-coordinate of one point to graph because whenx=8,x+2=10,the base of the common logarithm.

Graph of three functions. The parent function is y=log(x), with an asymptote at x=0. The first translation function y=5log(x+2) has an asymptote at x=-2. The second translation function y=log(x+2) has an asymptote at x=-2.
Figure 12.

The domain is(2,),the range is(,),and the vertical asymptote isx=2.[/hidden-answer]

Try It

Sketch a graph of the functionf(x)=3log(x2)+1.State the domain, range, and asymptote.

[reveal-answer q=”fs-id1165137437228″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165137437228″]Graph of f(x)=3log(x-2)+1 with an asymptote at x=2.

The domain is(2,),the range is(,),and the vertical asymptote isx=2.

[/hidden-answer]

Graphing Reflections of f(x) = logb(x)

When the parent functionf(x)=logb(x)is multiplied by1,the result is a reflection about the x-axis. When the input is multiplied by1,the result is a reflection about the y-axis. To visualize reflections, we restrictb>1,and observe the general graph of the parent functionf(x)=logb(x)alongside the reflection about the x-axis,g(x)=logb(x)and the reflection about the y-axis,h(x)=logb(x).

Figure 13.

Reflections of the Parent Function y = logb(x)

The functionf(x)=logb(x)

  • reflects the parent functiony=logb(x)about the x-axis.
  • has domain,(0,), range,(,), and vertical asymptote,x=0, which are unchanged from the parent function.

The functionf(x)=logb(x)

  • reflects the parent functiony=logb(x)about the y-axis.
  • has domain(,0).
  • has range,(,), and vertical asymptote,x=0, which are unchanged from the parent function.

Given a logarithmic function with the parent functionf(x)=logb(x), graph a translation.

If f(x)=logb(x) If f(x)=logb(x)
  1. Draw the vertical asymptote,x=0.
  1. Draw the vertical asymptote,x=0.
  1. Plot the x-intercept,(1,0).
  1. Plot the x-intercept,(1,0).
  1. Reflect the graph of the parent functionf(x)=logb(x)about the x-axis.
  1. Reflect the graph of the parent functionf(x)=logb(x)about the y-axis.
  1. Draw a smooth curve through the points.
  1. Draw a smooth curve through the points.
  1. State the domain,(0,), the range,(,), and the vertical asymptotex=0.
  1. State the domain,(,0), the range,(,), and the vertical asymptotex=0.

Graphing a Reflection of a Logarithmic Function

Sketch a graph off(x)=log(x)alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.

[reveal-answer q=”fs-id1165137836523″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165137836523″]

Before graphingf(x)=log(x),identify the behavior and key points for the graph.

  • Sinceb=10is greater than one, we know that the parent function is increasing. Since the input value is multiplied by1,fis a reflection of the parent graph about the y-axis. Thus,f(x)=log(x)will be decreasing asxmoves from negative infinity to zero, and the right tail of the graph will approach the vertical asymptotex=0.
  • The x-intercept is(1,0).
  • We draw and label the asymptote, plot and label the points, and draw a smooth curve through the points.
Graph of two functions. The parent function is y=log(x), with an asymptote at x=0 and labeled points at (1, 0), and (10, 0).The translation function f(x)=log(-x) has an asymptote at x=0 and labeled points at (-1, 0) and (-10, 1).
Figure 14.

The domain is(,0),the range is(,),and the vertical asymptote isx=0.[/hidden-answer]

Try It

Graphf(x)=log(x).State the domain, range, and asymptote.

[reveal-answer q=”fs-id1165137836698″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165137836698″]Graph of f(x)=-log(-x) with an asymptote at x=0.

The domain is(,0),the range is(,),and the vertical asymptote isx=0.

[/hidden-answer]

How To

Given a logarithmic equation, use a graphing calculator to approximate solutions.

  1. Press [Y=]. Enter the given logarithm equation or equations as Y1= and, if needed, Y2=.
  2. Press [GRAPH] to observe the graphs of the curves and use [WINDOW] to find an appropriate view of the graphs, including their point(s) of intersection.
  3. To find the value ofx, we compute the point of intersection. Press [2ND] then [CALC]. Select “intersect” and press [ENTER] three times. The point of intersection gives the value ofx,for the point(s) of intersection.

Approximating the Solution of a Logarithmic Equation

Solve4ln(x)+1=2ln(x1)graphically. Round to the nearest thousandth.

[reveal-answer q=”fs-id1165135193432″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165135193432″]Press [Y=] and enter4ln(x)+1next to Y1=. Then enter2ln(x1)next to Y2=. For a window, use the values 0 to 5 forxand –10 to 10 fory.Press [GRAPH]. The graphs should intersect somewhere a little to right ofx=1.

For a better approximation, press [2ND] then [CALC]. Select [5: intersect] and press [ENTER] three times. The x-coordinate of the point of intersection is displayed as 1.3385297. (Your answer may be different if you use a different window or use a different value for Guess?) So, to the nearest thousandth,x1.339.

[/hidden-answer]

Try It

Solve5log(x+2)=4log(x)graphically. Round to the nearest thousandth.

[reveal-answer q=”fs-id1165137817452″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165137817452″]

x3.049

[/hidden-answer]

Summarizing Translations of the Logarithmic Function

Now that we have worked with each type of translation for the logarithmic function, we can summarize each in (Figure) to arrive at the general equation for translating exponential functions.

 
Translations of the Parent Functiony=logb(x)
Translation Form
Shift

  • Horizontallycunits to the left
  • Verticallydunits up
y=logb(x+c)+d
Stretch and Compress

  • Stretch if|a|>1
  • Compression if|a|<1
y=alogb(x)
Reflect about the x-axis y=logb(x)
Reflect about the y-axis y=logb(x)
General equation for all translations y=alogb(x+c)+d

Translations of Logarithmic Functions

All translations of the parent logarithmic function,y=logb(x), have the form

f(x)=alogb(x+c)+d

where the parent function,y=logb(x),b>1,is

  • shifted vertically updunits.
  • shifted horizontally to the leftcunits.
  • stretched vertically by a factor of|a|if|a|>0.
  • compressed vertically by a factor of|a|if0<|a|<1.
  • reflected about the x-axis whena<0.

Forf(x)=log(x), the graph of the parent function is reflected about the y-axis.

Finding the Vertical Asymptote of a Logarithm Graph

What is the vertical asymptote off(x)=2log3(x+4)+5?

[reveal-answer q=”fs-id1165137572550″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165137572550″]

The vertical asymptote is atx=4.

[/hidden-answer]

Analysis

The coefficient, the base, and the upward translation do not affect the asymptote. The shift of the curve 4 units to the left shifts the vertical asymptote tox=4.

Try It

What is the vertical asymptote off(x)=3+ln(x1)?

[reveal-answer q=”fs-id1165135388504″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165135388504″]

x=1

[/hidden-answer]

Finding the Equation from a Graph

Find a possible equation for the common logarithmic function graphed in (Figure).

Graph of a logarithmic function with a vertical asymptote at x=-2, has been vertically reflected, and passes through the points (-1, 1) and (2, -1).
Figure 15.
[reveal-answer q=”fs-id1165135342977″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165135342977″]

This graph has a vertical asymptote atx=2and has been vertically reflected. We do not know yet the vertical shift or the vertical stretch. We know so far that the equation will have form:

f(x)=alog(x+2)+k

It appears the graph passes through the points(1,1)and(2,1).Substituting(1,1),

1=alog(1+2)+kSubstitute (1,1).1=alog(1)+kArithmetic.1=klog(1)=0.

Next, substituting in(2,1),

1=alog(2+2)+1Plug in (2,1).2=alog(4)Arithmetic. a=2log(4)Solve for a.

This gives us the equationf(x)=2log(4)log(x+2)+1.

[/hidden-answer]

Analysis

We can verify this answer by comparing the function values in (Figure) with the points on the graph in (Figure).

x −1 0 1 2 3
f(x) 1 0 −0.58496 −1 −1.3219
x 4 5 6 7 8
f(x) −1.5850 −1.8074 −2 −2.1699 −2.3219

Try It

Give the equation of the natural logarithm graphed in (Figure).

Graph of a logarithmic function with a vertical asymptote at x=-3, has been vertically stretched by 2, and passes through the points (-1, -1).
Figure 16.
[reveal-answer q=”fs-id1165137436435″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165137436435″]

f(x)=2ln(x+3)1

[/hidden-answer]

Is it possible to tell the domain and range and describe the end behavior of a function just by looking at the graph?

Yes, if we know the function is a general logarithmic function. For example, look at the graph in (Figure). The graph approachesx=3(or thereabouts) more and more closely, sox=3is, or is very close to, the vertical asymptote. It approaches from the right, so the domain is all points to the right,{x|x>3}.The range, as with all general logarithmic functions, is all real numbers. And we can see the end behavior because the graph goes down as it goes left and up as it goes right. The end behavior is that asx3+,f(x)and asx,f(x).

Access these online resources for additional instruction and practice with graphing logarithms.

Key Equations

General Form for the Translation of the Parent Logarithmic Function f(x)=logb(x) f(x)=alogb(x+c)+d

Key Concepts

  • To find the domain of a logarithmic function, set up an inequality showing the argument greater than zero, and solve forx.See (Figure) and (Figure)
  • The graph of the parent functionf(x)=logb(x)has an x-intercept at(1,0),domain(0,),range(,),vertical asymptotex=0,and
    • ifb>1,the function is increasing.
    • if[latex]\,0

    See (Figure).

  • The equationf(x)=logb(x+c)shifts the parent functiony=logb(x)horizontally
    • leftcunits ifc>0.
    • rightcunits ifc<0.

    See (Figure).

  • The equationf(x)=logb(x)+dshifts the parent functiony=logb(x)vertically
    • updunits ifd>0.
    • downdunits ifd<0.

    See (Figure).

  • For any constanta>0, the equationf(x)=alogb(x)
    • stretches the parent functiony=logb(x)vertically by a factor ofaif|a|>1.
    • compresses the parent functiony=logb(x)vertically by a factor ofaif|a|<1.

    See (Figure) and (Figure).

  • When the parent functiony=logb(x)is multiplied by1, the result is a reflection about the x-axis. When the input is multiplied by1, the result is a reflection about the y-axis.
    • The equationf(x)=logb(x)represents a reflection of the parent function about the x-axis.
    • The equationf(x)=logb(x)represents a reflection of the parent function about the y-axis.

    See (Figure).

    • A graphing calculator may be used to approximate solutions to some logarithmic equations See (Figure).
  • All translations of the logarithmic function can be summarized by the general equationf(x)=alogb(x+c)+d.See (Figure).
  • Given an equation with the general formf(x)=alogb(x+c)+d,we can identify the vertical asymptotex=cfor the transformation. See (Figure).
  • Using the general equationf(x)=alogb(x+c)+d,we can write the equation of a logarithmic function given its graph. See (Figure).

Section Exercises

Verbal

The inverse of every logarithmic function is an exponential function and vice-versa. What does this tell us about the relationship between the coordinates of the points on the graphs of each?

[reveal-answer q=”fs-id1165135582193″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165135582193″]

Since the functions are inverses, their graphs are mirror images about the liney=x.So for every point(a,b)on the graph of a logarithmic function, there is a corresponding point(b,a)on the graph of its inverse exponential function.

[/hidden-answer]

What type(s) of translation(s), if any, affect the range of a logarithmic function?

What type(s) of translation(s), if any, affect the domain of a logarithmic function?

[reveal-answer q=”917821″]Show Solution[/reveal-answer]
[hidden-answer a=”917821″]Shifting the function right or left and reflecting the function about the y-axis will affect its domain.[/hidden-answer]

Consider the general logarithmic functionf(x)=logb(x).Why can’txbe zero?

Does the graph of a general logarithmic function have a horizontal asymptote? Explain.

[reveal-answer q=”fs-id1165137459912″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165137459912″]

No. A horizontal asymptote would suggest a limit on the range, and the range of any logarithmic function in general form is all real numbers.

[/hidden-answer]

Algebraic

For the following exercises, state the domain and range of the function.

f(x)=log3(x+4)

h(x)=ln(12x)

[reveal-answer q=”fs-id1165135257219″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165135257219″]

Domain:(,12);Range:(,)

[/hidden-answer]

g(x)=log5(2x+9)2

h(x)=ln(4x+17)5

[reveal-answer q=”fs-id1165137641046″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165137641046″]

Domain:(174,);Range:(,)

[/hidden-answer]

f(x)=log2(123x)3

For the following exercises, state the domain and the vertical asymptote of the function.

f(x)=logb(x5)

[reveal-answer q=”fs-id1165137771436″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165137771436″]

Domain:(5,);Vertical asymptote:x=5

[/hidden-answer]

g(x)=ln(3x)

f(x)=log(3x+1)

[reveal-answer q=”fs-id1165135256140″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165135256140″]

Domain:(13,);Vertical asymptote:x=13

[/hidden-answer]

f(x)=3log(x)+2

g(x)=ln(3x+9)7

[reveal-answer q=”fs-id1165135532384″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165135532384″]

Domain:(3,);Vertical asymptote:x=3

[/hidden-answer]

For the following exercises, state the domain, vertical asymptote, and end behavior of the function.

f(x)=ln(2x)

f(x)=log(x37)

[reveal-answer q=”fs-id1165137715430″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165137715430″]Domain: (37,);[/hidden-answer]

h(x)=log(3x4)+3

g(x)=ln(2x+6)5

[reveal-answer q=”fs-id1165135358044″]Show Solution[/reveal-answer]

[hidden-answer a=”fs-id1165135358044″]

Domain: (3,); Vertical asymptote: x=3;[/hidden-answer]

f(x)=log3(155x)+6

For the following exercises, state the domain, range, and x– and y-intercepts, if they exist. If they do not exist, write DNE.

h(x)=log4(x1)+1

[reveal-answer q=”fs-id1165132983045″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165132983045″]

Domain:(1,);Range:(,);Vertical asymptote:x=1;x-intercept:(54,0);y-intercept: DNE

[/hidden-answer]

f(x)=log(5x+10)+3

g(x)=ln(x)2

[reveal-answer q=”fs-id1165137426893″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165137426893″]

Domain:(,0);Range:(,);Vertical asymptote:x=0;x-intercept:(e2,0);y-intercept: DNE

[/hidden-answer]

f(x)=log2(x+2)5

h(x)=3ln(x)9
[reveal-answer q=”fs-id1165135186502″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165135186502″]

Domain:(0,);Range:(,); Vertical asymptote: x=0;x-intercept:(e3,0);y-intercept: DNE

[/hidden-answer]

Graphical

For the following exercises, match each function in (Figure) with the letter corresponding to its graph.

Graph of five logarithmic functions.
Figure 17.

d(x)=log(x)

f(x)=ln(x)

[reveal-answer q=”fs-id1165134255558″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165134255558″]

B

[/hidden-answer]

g(x)=log2(x)

h(x)=log5(x)

[reveal-answer q=”fs-id1165135571660″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165135571660″]

C

[/hidden-answer]

j(x)=log25(x)

For the following exercises, match each function in (Figure) with the letter corresponding to its graph.

Graph of three logarithmic functions.
Figure 18.

f(x)=log13(x)

[reveal-answer q=”fs-id1165135206082″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165135206082″]B[/hidden-answer]

g(x)=log2(x)

h(x)=log34(x)

[reveal-answer q=”fs-id1165137637948″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165137637948″]

C

[/hidden-answer]

For the following exercises, sketch the graphs of each pair of functions on the same axis.

f(x)=log(x)andg(x)=10x

f(x)=log(x)andg(x)=log12(x)

[reveal-answer q=”809684″]Show Solution[/reveal-answer]
[hidden-answer a=”809684″]Graph of two functions, g(x) = log_(1/2)(x) in orange and f(x)=log(x) in blue.[/hidden-answer]

f(x)=log4(x)andg(x)=ln(x)

f(x)=exandg(x)=ln(x)

[reveal-answer q=”fs-id1165135496605″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165135496605″]Graph of two functions, g(x) = ln(1/2)(x) in orange and f(x)=e^(x) in blue.[/hidden-answer]

For the following exercises, match each function in (Figure) with the letter corresponding to its graph.

Graph of three logarithmic functions.
Figure 19.

f(x)=log4(x+2)

g(x)=log4(x+2)

[reveal-answer q=”478955″]Show Solution[/reveal-answer]
[hidden-answer a=”478955″]C[/hidden-answer]

h(x)=log4(x+2)

For the following exercises, sketch the graph of the indicated function.

f(x)=log2(x+2)

[reveal-answer q=”916976″]Show Solution[/reveal-answer]
[hidden-answer a=”916976″]Graph of f(x)=log_2(x+2).[/hidden-answer]

f(x)=2log(x)

f(x)=ln(x)

[reveal-answer q=”fs-id1165134032402″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165134032402″]Graph of f(x)=ln(-x).[/hidden-answer]

g(x)=log(4x+16)+4

g(x)=log(63x)+1

[reveal-answer q=”fs-id1165137705397″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165137705397″]Graph of g(x)=log(6-3x)+1.[/hidden-answer]

h(x)=12ln(x+1)3

For the following exercises, write a logarithmic equation corresponding to the graph shown.

Usey=log2(x)as the parent function.

The graph y=log_2(x) has been reflected over the y-axis and shifted to the right by 1.

[reveal-answer q=”fs-id1165134360851″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165134360851″]

f(x)=log2((x1))

[/hidden-answer]

Usef(x)=log3(x)as the parent function.

The graph y=log_3(x) has been reflected over the x-axis, vertically stretched by 3, and shifted to the left by 4.

Usef(x)=log4(x)as the parent function.

The graph y=log_4(x) has been vertically stretched by 3, and shifted to the left by 2.

[reveal-answer q=”fs-id1165134086084″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165134086084″]

f(x)=3log4(x+2)

[/hidden-answer]

Usef(x)=log5(x)as the parent function.

The graph y=log_3(x) has been reflected over the x-axis and y-axis, vertically stretched by 2, and shifted to the right by 5.

Technology

For the following exercises, use a graphing calculator to find approximate solutions to each equation.

log(x1)+2=ln(x1)+2

[reveal-answer q=”fs-id1165137674291″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165137674291″]

x=2

[/hidden-answer]

log(2x3)+2=log(2x3)+5

ln(x2)=ln(x+1)

[reveal-answer q=”fs-id1165135337765″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165135337765″]

x2.303

[/hidden-answer]

2ln(5x+1)=12ln(5x)+1

13log(1x)=log(x+1)+13

[reveal-answer q=”fs-id1165135174505″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165135174505″]

x0.472

[/hidden-answer]

Extensions

Letbbe any positive real number such thatb1.What mustlogb1be equal to? Verify the result.

Explore and discuss the graphs off(x)=log12(x)andg(x)=log2(x).Make a conjecture based on the result.

[reveal-answer q=”fs-id1165134151930″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165134151930″]

The graphs off(x)=log12(x)andg(x)=log2(x)appear to be the same; Conjecture: for any positive baseb1,logb(x)=log1b(x).

[/hidden-answer]

Prove the conjecture made in the previous exercise.

What is the domain of the functionf(x)=ln(x+2x4)?Discuss the result.

[reveal-answer q=”fs-id1165135593381″]Show Solution[/reveal-answer]
[hidden-answer a=”fs-id1165135593381″]

Recall that the argument of a logarithmic function must be positive, so we determine wherex+2x4>0. From the graph of the functionf(x)=x+2x4, note that the graph lies above the x-axis on the interval(,2)and again to the right of the vertical asymptote, that is(4,).Therefore, the domain is(,2)(4,).

[/hidden-answer]

Use properties of exponents to find the x-intercepts of the functionf(x)=log(x2+4x+4)algebraically. Show the steps for solving, and then verify the result by graphing the function.

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PPSC MAT 1420: Algebra and Trigonometry by OpenStax is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.

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