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2B. Exponential and Logarithmic Functions
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AP Precalculus

2.11 Logarithmic functions

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Lesson

Introduction

Learning objective

  • 2.11.A Identify key characteristics of logarithmic functions.

Logarithmic functions

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When the base b>1, the graph of the parent logarithmic function is a rising curve that increases at a decreasing rate. Note, that even though there is a decreasing rate of increase, there is no limit on the function's range.

  • The graph is a strictly increasing function

  • The domain is \left(0, \infty\right)

  • The range is \left(-\infty, \infty\right)

  • The x-intercept is at \left(1,\, 0\right)

  • The vertical asymptote is x=0

When we examine the end behavior on the left side of the graph, we can see there is now a vertical asymptote at x = 0. As the x-values approach 0 from the positive side, x\to 0^{+}, f\left(x\right)\to -\infty.

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When 0<b<1, the graph is a falling curve that decreases at a decreasing rate.

  • The graph is a strictly decreasing function

  • The domain is \left(0, \infty\right)

  • The range is \left(-\infty, \infty\right)

  • The x-intercept is at \left(1,\, 0\right)

  • The vertical asymptote is x=0

Logarithmic functions can be dilated, reflected, and translated in a similar way to other functions.

The logarithmic parent function f\left(x\right)=\log_b\left(x\right) can be transformed to f\left(x\right)=a\log_b\left[c\left(x-h\right)\right]+k

  • If a<0, the basic curve is reflected across the x-axis

  • The graph is vertically stretched or compressed by a factor of a

  • If c<0, the basic curve is reflected across the y-axis

  • The graph is horizontally stretched or compressed by a factor of c

  • The graph is translated horizontally by h units

  • The graph is translated vertically by k units

Examples

Example 1

Consider the function y=4\log_{2}(x-7).

a

Solve for the x-coordinate of the x-intercept.

Worked Solution
Create a strategy

We substitute y=0 in the function and solve for x.

Apply the idea
\displaystyle 0\displaystyle =\displaystyle 4\log_{2}(x-7)Substitute y=0
\displaystyle 0\displaystyle =\displaystyle \log_{2}(x-7)Divide both sides by 4
\displaystyle 2^{0}\displaystyle =\displaystyle x-7Transform the equation from logarithmic to exponential form
\displaystyle 1\displaystyle =\displaystyle x-7Simplify 2^{0}
\displaystyle x\displaystyle =\displaystyle 8Add 7 to both sides

So, the x-coordinate of the x-intercept is 8.

b

State the equation of the vertical asymptote.

Worked Solution
Create a strategy

The vertical asymptote of a logarithmic function y = a\log_b(x - h) + k occurs at x = h, as it's the vertical line that the graph of the function approaches but never crosses.

Apply the idea

For the function y=4\log_2(x-7), the vertical asymptote is at x = 7.

c

Sketch the graph of the function.

Worked Solution
Create a strategy

The graph of this function will pass through the x-intercept at (8,\, 0) and approach but never cross the vertical asymptote at x = 7.

Apply the idea
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Idea summary

The graph of the parent logarithmic function has the following characteristics:

  • The domain is \left(0, \infty\right)

  • The range is \left(-\infty, \infty\right)

  • The x-intercept is at \left(1,\, 0\right)

  • The vertical asymptote is x=0

When b>1, the function is strictly increasing. When 0<b<1, the function is strictly decreasing.

We can use the inverse relationship between logarithmic and exponential functions to find key points with which to sketch the graph of a logarithmic function.

Transformations of logarithmic graphs

A logarithmic graph can be vertically translated by increasing or decreasing the y-values by a constant number. So to translate y=\log_{2}x up by k units gives us y=\log_{2}x + k.

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This graph shows y=\log_{2}x translated vertically up by 2 to get y=\log_{2}x + 2, and down by 2 to get y=\log_{2}x -2.

Similarly, a logarithmic graphh can be horizontally translated by increasing or decreasing the x-values by a constant number. However, the x-value together with the translation must both be in the logarithm. That is, to translate y=\log_{2}x to the left by h units we get y=\log_{2}(x+h).

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This graph shows y=\log_{2}x translated horizontally to the left by 2 to get y=\log_{2}(x+2), and to the right by 2 to get y=\log_{2}(x-2).

A logarithmic graph can be vertically scaled by multiplying every y-value by a constant number. So to expand the logarithmic graph y=\log_{2}x by a scale factor of a we get y=a \log_{2}x. We can compress an exponential graph by dividing by the scale factor instead.

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This graph shows y=\log_{2}x vertically expanded by a scale factor of 2 to get y=2\log_{2}x and compressed by a scale factor of 2 to gety=\dfrac{1}{2}\log_{2}x.

We can vertically reflect a logarithmic graph about the x-axis by taking the negative of the y-values. So to reflect y=\log_{2}x about the x-axis gives us y=-\log_{2}x.

We can similarly horizontally reflect a logarithmic graph about the y-axis by taking the negative of the x-values. So to reflect y=\log_{2}x about the y-axis gives us y=\log_{2}(-x).

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This graph shows y=\log_{2}x horizontally reflected or graph about the y-axis to get y=-\log_{2}x and vertically reflected or graph about the x-axis to get y=\log_{2}(-x).

Exploration

Use the following applet to explore transformations of the graph of a logarithmic function by dragging the sliders.

Loading interactive...

Changing B changes the steepness of the graph. Changing A changes the steepness of the graph and negative values of A flip the curve horizontally. Changing h shifts the curve horizontally, and changing k shifts the curve vertically.

Examples

Example 2

A graph of the function y = \log_{3} x is shown below.

A graph of the function y = \log_{3} x + 3 can be obtained from the original graph by transforming it in some way.

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a

Complete the table of values below for y=\log_{3} x:

x\dfrac{1}{3}139
\log_3 x
Worked Solution
Create a strategy

Substitute each of the x-values in the table into the equation and evaluate.

Apply the idea

For x=\dfrac{1}{3}=0.3333:

\displaystyle y\displaystyle =\displaystyle \log_{3} x
\displaystyle =\displaystyle \log_{3} 0.3333Substitute x=0.3333
\displaystyle =\displaystyle \dfrac{\log_{10} 0.3333}{\log_{10} 3}Use the change of base law
\displaystyle =\displaystyle -1Evaluate

Similarly, by substituting the remaining x-values into \log_{3} x, we get:

x\dfrac{1}{3}139
\log_3 x-1012
b

Now complete the table of values below for y=\log_{3} x + 3:

x\dfrac{1}{3}139
\log_3 x +3
Worked Solution
Create a strategy

Add 3 to each of the resulting y-values of the table for y=\log_{3} x.

Apply the idea

In part (a) we have the tables of values for y=\log_{3} x, such as if x=\dfrac{1}{3}, \log_{3} x=-1.

x\dfrac{1}{3}139
\log_3 x-1012

So for y=\log_{3} x + 3, \, x=\dfrac{1}{3} we have:

\displaystyle y\displaystyle =\displaystyle \log_{3} x + 3
\displaystyle =\displaystyle \log_{3} \dfrac{1}{3} + 3Substitute x=\dfrac{1}{3}
\displaystyle =\displaystyle -1+3Substitute \log_{3} \dfrac{1}{3}=-1
\displaystyle =\displaystyle 2Evaluate

Similarly, by substituting the remaining x-values into \log_{3} x +3, we get:

x\dfrac{1}{3}139
\log_3 x2345
c

Which of the following is a graph of y=\log_{3} x +3?

A
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B
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C
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D
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Worked Solution
Create a strategy

Plot the points in the table of values and draw the curve passing through each plotted point.

Apply the idea
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From the table of values in part (b) we have some the ordered pairs of points to be plotted on the coordinate plane: (1,3), (3,4) , and (9,5).

This curve of the equation \log_{4} x + 3 must pass through each of the plotted points.

So option B is the correct answer.

d

Which features of the graph are unchanged after it has been translated 3 units upwards?

A
The range.
B
The general shape of the graph.
C
The vertical asymptote.
D
The x-intercept.
Worked Solution
Create a strategy

Compare the graphs of the two equations.

Apply the idea
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Comparing the two graphs of the wo equations y=\log_{3} and y=\log_{3} x +3, we can see from the graph that the range, the general shape of the graph and the vertical asypmtote are unchanged after it has been translated 3 units upward.

So options A, B and C are the correct answers.

Example 3

Given the graph of y=\log_{6} (-x) , draw the graph of y=5\log_{6} (-x) on the same plane.

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Worked Solution
Create a strategy

Vertically expand the given graph by a scale factor 5.

Apply the idea
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The graph of y=5 \log_{6} (-x) can be obtained from the graph of y=5 \log_{6} (-x) by vertically expanding it with a scale factor 5.

The transformation does not change the location of the asymptote or the x-intercept, we will only need to change the location of the a point on the graph.

Let us look at the point (-6,1) on the original graph. The new y-coordinate of this point when it is multiplied by 5 is (-6,5).

So we have the graph of y=5 \log_{6} (-x).

Idea summary

Logarithmic graphs can be transformed in the following ways (starting with the logarithmic graph defined by y=\log_{2} x):

  • Vertically translated by k units: y=\log_{2} x + k

  • Horizontally translated by h units: y=\log_{2} (x-h)

  • Vertically scaled by a scale factor of a: y=a\log_{2} x

  • Vertically reflected about the x-axis: y=-\log_{2} x

  • Horizontally reflected about the y-axis: y=\log_{2} (-x)

Outcomes

2.11.A

Identify key characteristics of logarithmic functions.

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