Section 4.4Graphs of Logarithmic Functions

Corequisite Skills review (optional warm-up)

Learning Objectives

• Find the domain and range of a relation and a function. (IA 3.5.1)
• Graph Logarithmic functions. (IA 10.3.3)

Objective 1: Find the domain and range of a relation and a function. (IA 3.5.1)

Vocabulary

Warm-up Example 1

Find the domain and range of a relation and a function.

• ⓐ
• ⓑ
• ⓒ Find the domain of the function \(f(x)=\frac{5}{x-2}\)

  Find the domain of the function \(f(x)=\frac{5}{x-2}\)

• ⓓ Find the domain of the function \(f(x)={log}_{2}(x-5)\) .

  Find the domain of the function \(f(x)={log}_{2}(x-5)\) .

Show solution

• ⓐ The set of points on the graph is \({(-4,-2),(-2,-1),(-1,1),(1,2)}\)
  The Domain is the set of all x-coordinates: \({-4,-2,-1,1}\)
  The Range is the set of all y-coordinates: \({-2,-1,1}\)
  Notice that even though y-coodinate of 1 appears twice, we only list it once.

  The set of points on the graph is \({(-4,-2),(-2,-1),(-1,1),(1,2)}\)
  The Domain is the set of all x-coordinates: \({-4,-2,-1,1}\)
  The Range is the set of all y-coordinates: \({-2,-1,1}\)
  Notice that even though y-coodinate of 1 appears twice, we only list it once.

• ⓑ Domain: \((-\infty,\infty)\)
  Range: \([-2,\infty)\)
  Notice that \(-2\) is included because the point \((3,-2)\) is on the graph of a function.

  Domain: \((-\infty,\infty)\)
  Range: \([-2,\infty)\)
  Notice that \(-2\) is included because the point \((3,-2)\) is on the graph of a function.

• ⓒ A function is not defined when the denominator is zero. We need to set the denominator equal zero and exclude this value(s) from the domain.
  \(x-2=0,x=2,\) Domain \((-\infty,2)∪(2,\infty)\)
  Notice that 2 is excluded from the domain because the function is not defined at \(x=2\)

  A function is not defined when the denominator is zero. We need to set the denominator equal zero and exclude this value(s) from the domain.
  \(x-2=0,x=2,\) Domain \((-\infty,2)∪(2,\infty)\)
  Notice that 2 is excluded from the domain because the function is not defined at \(x=2\)

• ⓓ From the definition of the logarithmic function \(f(x)={log}_{a}x\,\) we know that \(x>0\)
  To find domain of \(f(x)={log}_{2}(x-5\) , we need to set up and solve inequality.
  \(x-5>0\) ,
  \(x>5)\) Domain: \((5,\infty)\)

  From the definition of the logarithmic function \(f(x)={log}_{a}x\,\) we know that \(x>0\)
  To find domain of \(f(x)={log}_{2}(x-5\) , we need to set up and solve inequality.
  \(x-5>0\) ,
  \(x>5)\) Domain: \((5,\infty)\)

Practice Makes Perfect

Find the domain and range of a relation and a function.

P1

Find the domain and range of a relation.

P2

Find the domain and the range of the function graphed. Use interval notation.

P3

Find the domain of the function \(f(x)={log}_{2}(x+4)\) . Notice: this is the same function that was graphed in question 2.

Objective 2: Graph Logarithmic functions. (IA 10.3.3)

To graph a logarithmic function \(y={log}_{a}x\) , it is easiest to convert the equation to its exponential form, \(x={a}^{y}\) . Generally, when we look for ordered pairs for the graph of a function, we usually choose an x-value and then determine its corresponding y-value. In this case you may find it easier to choose y-values and then determine its corresponding x-value.

Warm-up Example 2

Graph Logarithmic functions.

Graph \(y={\text{log}}_{2}x.\)

Show solution

To graph the function, we will first rewrite the logarithmic equation, \(y={\text{log}}_{2}x,\) in exponential form, \({2}^{y}=x.\)

We will use point plotting to graph the function. It will be easier to start with values of y and then get x.

Table 1

\(y\)	\({2}^{y}=x\)	\((x,y)\)
\(-2\)	\({2}^{-2}=\frac{1}{{2}^{2}}=\frac{1}{4}\)	\((\frac{1}{4},2)\)
\(-1\)	\({2}^{-1}=\frac{1}{{2}^{1}}=\frac{1}{2}\)	\((\frac{1}{2},-1)\)
0	\({2}^{0}=1\)	\((1,0)\)
1	\({2}^{1}=2\)	\((2,1)\)
2	\({2}^{2}=4\)	\((4,2)\)
3	\({2}^{3}=8\)	\((8,3)\)

Practice Makes Perfect

Graph Logarithmic functions

P4

Graph \(y={\text{log}}_{3}x\) and \(y={\text{log}}_{5}x\) in the same coordinate system.

Table 2

\(y\)	\({3}^{y}=x\)	\((x,y)\)
\(y\)	\({5}^{y}=x\)	\((x,y)\)

P5

Graph \(y={log}_{1/3}x\)

Table 3

\(y\)	\({(\frac{1}{3})}^{y}=x\)	\((x,y)\)

P6

Do the graphs of \(y={log}_{2}x\) , \(y={log}_{3}x\) , and \(y={log}_{5}x\) have the shape we expect from a logarithmic function where \(a>0\) ? (Remember a is the base of the log function)

P7

Is there a point they all share? Why does this make sense?

P8

Do they all have a point \((a,1)\) ? Why does this make sense?

P9

Do they all have a point \((\frac{1}{a},-1)\) ? Why does this make sense?

P10

Do they all have the same vertical asymptote? What is the equation of the vertical asymptote?

P11

Do they all have the same domain? Write the domain in the interval notation.

P12

Do they all have the same range? Write the range in the interval notation.

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 \(\text{\$}2500\) in an account that offers an annual interest rate of \(5\%,\) compounded continuously. We already know that the balance in our account for any year \(t\) can be found with the equation \(A=2500{e}^{0.05t}.\)

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 1 shows this point on the logarithmic graph.

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 as \(y={b}^{x}\) for any real number \(x\) and constant \(b>0,\) \(b\ne1,\) where

• The domain of \(y\) is \((-\infty,\infty).\)
• The range of \(y\) is \((0,\infty).\)

In the last section we learned that the logarithmic function \(y={log}_{b}(x)\) is the inverse of the exponential function \(y={b}^{x}.\) So, as inverse functions:

• The domain of \(y={log}_{b}(x)\) is the range of \(y={b}^{x}:\) \((0,\infty).\)
• The range of \(y={log}_{b}(x)\) is the domain of \(y={b}^{x}:\) \((-\infty,\infty).\)

Transformations of the parent function \(y={log}_{b}(x)\) 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.

In Graphs of Exponential Functions we saw that certain transformations can change the range of \(y={b}^{x}.\) Similarly, applying transformations to the parent function \(y={log}_{b}(x)\) 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, consider \(f(x)={log}_{4}(2x-3).\) This function is defined for any values of \(x\) such that the argument, in this case \(2x-3,\) is greater than zero. To find the domain, we set up an inequality and solve for \(x:\)

\[\begin{array}{ll}2x-3>0 & \text{Show the argument greater than zero}. \\ \ \ \ \ \ \ \ \ \ \ 2x>3 & \text{Add 3}. \\ \ \ \ \ \ \ \ \ \ \ \ \ \ x>1.5\begin{array}{llll} & & & \end{array} & \text{Divide by 2}.\end{array}\]

In interval notation, the domain of \(f(x)={log}_{4}(2x-3)\) is \((1.5,\infty).\)

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 function \(y={log}_{b}(x)\) along with all its transformations: shifts, stretches, compressions, and reflections.

We begin with the parent function \(y={log}_{b}(x).\) Because every logarithmic function of this form is the inverse of an exponential function with the form \(y={b}^{x},\) their graphs will be reflections of each other across the line \(y=x.\) To illustrate this, we can observe the relationship between the input and output values of \(y={2}^{x}\) and its equivalent \(x={log}_{2}(y)\) in Table 4.

Using the inputs and outputs from Table 4, we can build another table to observe the relationship between points on the graphs of the inverse functions \(f(x)={2}^{x}\) and \(g(x)={log}_{2}(x).\) See Table 5.

As we’d expect, the x- and y-coordinates are reversed for the inverse functions. Figure 2 shows the graph of \(f\) and \(g.\)

Observe the following from the graph:

• \(f(x)={2}^{x}\) has a y-intercept at \((0,1)\) and \(g(x)={log}_{2}(x)\) has an x- intercept at \((1,0).\)
• The domain of \(f(x)={2}^{x},\) \((-\infty,\infty),\) is the same as the range of \(g(x)={log}_{2}(x).\)
• The range of \(f(x)={2}^{x},\) \((0,\infty),\) is the same as the domain of \(g(x)={log}_{2}(x).\)

Try It #3

Graph \(f(x)={log}_{\frac{1}{5}}(x).\) State the domain, range, and asymptote.

Show answer

The domain is \((0,\infty),\) the range is \((-\infty,\infty),\) and the vertical asymptote is \(x=0.\)

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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 function \(y={log}_{b}(x)\) without loss of shape.

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

When a constant \(c\) is added to the input of the parent function \(f(x)=lo{g}_{b}(x),\) the result is a horizontal shift \(c\) units in the opposite direction of the sign on \(c.\) To visualize horizontal shifts, we can observe the general graph of the parent function \(f(x)={log}_{b}(x)\) and for \(c>0\) alongside the shift left, \(g(x)={log}_{b}(x+c),\) and the shift right, \(h(x)={log}_{b}(x-c).\) See Figure 6.

Try It #4

Sketch a graph of \(f(x)={log}_{3}(x+4)\) alongside its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.

Show answer

The domain is \((-4,\infty),\) the range \((-\infty,\infty),\) and the asymptote \(x=-4.\)

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Graphing a Vertical Shift of f(x) = log_b(x)

When a constant \(d\) is added to the parent function \(f(x)={log}_{b}(x),\) the result is a vertical shift \(d\) units in the direction of the sign on \(d.\) To visualize vertical shifts, we can observe the general graph of the parent function \(f(x)={log}_{b}(x)\) alongside the shift up, \(g(x)={log}_{b}(x)+d\) and the shift down, \(h(x)={log}_{b}(x)-d.\) See Figure 8.

Try It #5

Sketch a graph of \(f(x)={log}_{2}(x)+2\) alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.

Show answer

The domain is \((0,\infty),\) the range is \((-\infty,\infty),\) and the vertical asymptote is \(x=0.\)

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Graphing Stretches and Compressions of f(x) = log_b(x)

When the parent function \(f(x)={log}_{b}(x)\) is multiplied by a constant \(a>0,\) the result is a vertical stretch or compression of the original graph. To visualize stretches and compressions, we set \(a>1\) and observe the general graph of the parent function \(f(x)={log}_{b}(x)\) alongside the vertical stretch, \(g(x)=a{log}_{b}(x)\) and the vertical compression, \(h(x)=\frac{1}{a}{log}_{b}(x).\) See Figure 10.

Try It #6

Sketch a graph of \(f(x)=\frac{1}{2}\ {log}_{4}(x)\) alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.

Show answer

The domain is \((0,\infty),\) the range is \((-\infty,\infty),\) and the vertical asymptote is \(x=0.\)

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Try It #7

Sketch a graph of the function \(f(x)=3log(x-2)+1.\) State the domain, range, and asymptote.

Show answer

The domain is \((2,\infty),\) the range is \((-\infty,\infty),\) and the vertical asymptote is \(x=2.\)

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

When the parent function \(f(x)={log}_{b}(x)\) is multiplied by \(-1,\) the result is a reflection about the x-axis. When the input is multiplied by \(-1,\) the result is a reflection about the y-axis. To visualize reflections, we restrict \(b>1,\) and observe the general graph of the parent function \(f(x)={log}_{b}(x)\) alongside the reflection about the x-axis, \(g(x)={-log}_{b}(x)\) and the reflection about the y-axis, \(h(x)={log}_{b}(-x).\)

Try It #8

Graph \(f(x)=-log(-x).\) State the domain, range, and asymptote.

Show answer

The domain is \((-\infty,0),\) the range is \((-\infty,\infty),\) and the vertical asymptote is \(x=0.\)

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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 Table 7 to arrive at the general equation for translating exponential functions.

Key Equations

Key Concepts

• To find the domain of a logarithmic function, set up an inequality showing the argument greater than zero, and solve for \(x.\) See Example 1 and Example 2.
• The graph of the parent function \(f(x)={log}_{b}(x)\) has an x-intercept at \((1,0),\) domain \((0,\infty),\) range \((-\infty,\infty),\) vertical asymptote \(x=0.\)
  • if \(b>1,\) the function is increasing.
  • if \(0 < b < 1,\) the function is decreasing.
• The equation \(f(x)={log}_{b}(x+c)\) shifts the parent function \(y={log}_{b}(x)\) horizontally. See Example 4.
  • left \(c\) units if \(c>0.\)
  • right \(c\) units if \(c<0.\)
• The equation \(f(x)={log}_{b}(x)+d\) shifts the parent function \(y={log}_{b}(x)\) vertically. See Example 5.
  • up \(d\) units if \(d>0.\)
  • down \(d\) units if \(d<0.\)
• For any constant \(a>0,\) the equation \(f(x)=a{log}_{b}(x)\) See Example 6 and Example 7.
  • stretches the parent function \(y={log}_{b}(x)\) vertically by a factor of \(a\) if \(|a|>1.\)
  • compresses the parent function \(y={log}_{b}(x)\) vertically by a factor of \(a\) if \(|a|<1.\)
• When the parent function \(y={log}_{b}(x)\) is multiplied by \(-1,\) the result is a reflection about the x-axis. When the input is multiplied by \(-1,\) the result is a reflection about the y-axis. See Example 8.
  • The equation \(f(x)=-{log}_{b}(x)\) represents a reflection of the parent function about the x-axis.
  • The equation \(f(x)={log}_{b}(-x)\) represents a reflection of the parent function about the y-axis.
  • A graphing calculator may be used to approximate solutions to some logarithmic equations. See Example 9.
• All translations of the logarithmic function can be summarized by the general equation \(f(x)=a{log}_{b}(x+c)+d.\)
• Given an equation with the general form \(f(x)=a{log}_{b}(x+c)+d,\) we can identify the vertical asymptote \(x=-c\) for the transformation. See Example 10.
• Using the general equation \(f(x)=a{log}_{b}(x+c)+d,\) we can write the equation of a logarithmic function given its graph. See Example 11.

Section Exercises

Verbal

1

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?

Show answer

Since the functions are inverses, their graphs are mirror images about the line \(y=x.\) 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.

2

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

3

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

Show answer

Shifting the function right or left and reflecting the function about the y-axis will affect its domain.

4

Consider the general logarithmic function \(f(x)={log}_{b}(x).\) Why can’t \(x\) be zero?

5

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

Show answer

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.

Algebraic

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

6

\(f(x)={log}_{3}(x+4)\)

7

\(h(x)=ln(\frac{1}{2}-x)\)

Show answer

Domain: \((-\infty,\frac{1}{2});\) Range: \((-\infty,\infty)\)

8

\(g(x)={log}_{5}(2x+9)-2\)

9

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

Show answer

Domain: \((-\frac{17}{4},\infty);\) Range: \((-\infty,\infty)\)

10

\(f(x)={log}_{2}(12-3x)-3\)

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

11

\(f(x)={log}_{b}(x-5)\)

Show answer

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

12

\(g(x)=ln(3-x)\)

13

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

Show answer

Domain: \((-\frac{1}{3},\infty);\) Vertical asymptote: \(x=-\frac{1}{3}\)

14

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

15

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

Show answer

Domain: \((-3,\infty);\) Vertical asymptote: \(x=-3\)

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

16

\(f(x)=ln(2-x)\)

17

\(f(x)=log(x-\frac{3}{7})\)

Show answer

Domain: \((\frac{3}{7},\infty)\) ;
Vertical asymptote: \(x=\frac{3}{7}\) ; End behavior: as \(x\to{(\frac{3}{7})}^{+},f(x)\to-\infty\) and as \(x\to\infty,f(x)\to\infty\)

18

\(h(x)=-log(3x-4)+3\)

19

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

Show answer

Domain: \((-3,\infty)\) ; Vertical asymptote: \(x=-3\) ;
End behavior: as \(x\to-{3}^{+}\) , \(f(x)\to-\infty\) and as \(x\to\infty\) , \(f(x)\to\infty\)

20

\(f(x)={log}_{3}(15-5x)+6\)

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

21

\(h(x)={log}_{4}(x-1)+1\)

Show answer

Domain: \((1,\infty);\) Range: \((-\infty,\infty);\) Vertical asymptote: \(x=1;\) x-intercept: \((\frac{5}{4},0);\) y-intercept: DNE

22

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

23

\(g(x)=ln(-x)-2\)

Show answer

Domain: \((-\infty,0);\) Range: \((-\infty,\infty);\) Vertical asymptote: \(x=0;\) x-intercept: \((-{e}^{2},0);\) y-intercept: DNE

24

\(f(x)={log}_{2}(x+2)-5\)

25

\(h(x)=3ln(x)-9\)

Show answer

Domain: \((0,\infty);\) Range: \((-\infty,\infty);\) Vertical asymptote: \(x=0;\) x-intercept: \(({e}^{3},0);\) y-intercept: DNE

Graphical

For the following exercises, match each function in Figure 17 with the letter corresponding to its graph.

26

\(d(x)=log(x)\)

27

\(f(x)=ln(x)\)

Show answer

B

28

\(g(x)={log}_{2}(x)\)

29

\(h(x)={log}_{5}(x)\)

Show answer

C

30

\(j(x)={log}_{25}(x)\)

For the following exercises, match each function in Figure 18 with the letter corresponding to its graph.

31

\(f(x)={log}_{\frac{1}{3}}(x)\)

Show answer

B

32

\(g(x)={log}_{2}(x)\)

33

\(h(x)={log}_{\frac{3}{4}}(x)\)

Show answer

C

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

34

\(f(x)=log(x)\) and \(g(x)={10}^{x}\)

35

\(f(x)=log(x)\) and \(g(x)={log}_{\frac{1}{2}}(x)\)

Show answer

36

\(f(x)={log}_{4}(x)\) and \(g(x)=ln(x)\)

37

\(f(x)={e}^{x}\) and \(g(x)=ln(x)\)

Show answer

For the following exercises, match each function in Figure 19 with the letter corresponding to its graph.

38

\(f(x)={log}_{4}(-x+2)\)

39

\(g(x)=-{log}_{4}(x+2)\)

Show answer

C

40

\(h(x)={log}_{4}(x+2)\)

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

41

\(f(x)={log}_{2}(x+2)\)

Show answer

42

\(f(x)=2log(x)\)

43

\(f(x)=ln(-x)\)

Show answer

44

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

45

\(g(x)=log(6-3x)+1\)

Show answer

46

\(h(x)=-\frac{1}{2}ln(x+1)-3\)

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

47

Use \(y={log}_{2}(x)\) as the parent function.

Show answer

\(f(x)={log}_{2}(-(x-1))\)

48

Use \(f(x)={log}_{3}(x)\) as the parent function.

49

Use \(f(x)={log}_{4}(x)\) as the parent function.

Show answer

\(f(x)=3{log}_{4}(x+2)\)

50

Use \(f(x)={log}_{5}(x)\) as the parent function.

Technology

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

51

\(log(x-1)+2=ln(x-1)+2\)

Show answer

\(x=2\)

52

\(log(2x-3)+2=-log(2x-3)+5\)

53

\(ln(x-2)=-ln(x+1)\)

Show answer

\(x\approx\text{2}\text{.303}\)

54

\(2ln(5x+1)=\frac{1}{2}ln(-5x)+1\)

55

\(\frac{1}{3}log(1-x)=log(x+1)+\frac{1}{3}\)

Show answer

\(x\approx-0.472\)

Extensions

56

Let \(b\) be any positive real number such that \(b\ne1.\) What must \({log}_{b}1\) be equal to? Verify the result.

57

Explore and discuss the graphs of \(f(x)={log}_{\frac{1}{2}}(x)\) and \(g(x)=-{log}_{2}(x).\) Make a conjecture based on the result.

Show answer

The graphs of \(f(x)={log}_{\frac{1}{2}}(x)\) and \(g(x)=-{log}_{2}(x)\) appear to be the same; Conjecture: for any positive base \(b\ne1,\) \({log}_{b}(x)=-{log}_{\frac{1}{b}}(x).\)

58

Prove the conjecture made in the previous exercise.

59

What is the domain of the function \(f(x)=ln(\frac{x+2}{x-4})?\) Discuss the result.

Show answer

Recall that the argument of a logarithmic function must be positive, so we determine where \(\frac{x+2}{x-4}>0\) . From the graph of the function \(f(x)=\frac{x+2}{x-4},\) note that the graph lies above the x-axis on the interval \((-\infty,-2)\) and again to the right of the vertical asymptote, that is \((4,\infty).\) Therefore, the domain is \((-\infty,-2)∪(4,\infty).\)

60

Use properties of exponents to find the x-intercepts of the function \(f(x)=log({x}^{2}+4x+4)\) algebraically. Show the steps for solving, and then verify the result by graphing the function.