Previously, 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 $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 [latex]A=2500{e}^{0.05t}[/latex].

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? The graph below shows this point.

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.

Characteristics of Graphs of Logarithmic Functions

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 [latex]y={b}^{x}[/latex] for any real number x and constant [latex]b>0[/latex], [latex]b\ne 1[/latex], where

• The domain of y is [latex]\left(-\infty ,\infty \right)[/latex].
• The range of y is [latex]\left(0,\infty \right)[/latex].

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

• The domain of [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] is the range of [latex]y={b}^{x}[/latex]: [latex]\left(0,\infty \right)[/latex].
• The range of [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] is the domain of [latex]y={b}^{x}[/latex]: [latex]\left(-\infty ,\infty \right)[/latex].

Transformations of the parent function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] 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.

Previously we saw that certain transformations can change the range of [latex]y={b}^{x}[/latex]. Similarly, applying transformations to the parent function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] can change the domain. Therefore, when finding the domain of a logarithmic function, it is important to remember that the domain consists only of positive real numbers. That is, the value you are applying the logarithmic function to, also known as the argument of the logarithmic function, must be greater than zero.

For example, consider [latex]f\left(x\right)={\mathrm{log}}_{4}\left(2x - 3\right)[/latex]. This function is defined for any values of x such that the argument, in this case [latex]2x - 3[/latex], is greater than zero. To find the domain, we set up an inequality and solve for x:

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

In interval notation, the domain of [latex]f\left(x\right)={\mathrm{log}}_{4}\left(2x - 3\right)[/latex] is [latex]\left(1.5,\infty \right)[/latex].

Graphing a Logarithmic Function Using a Table of Values

Now that we have a feel for the set of values for which a logarithmic function is defined, we move on to graphing logarithmic functions. The family of logarithmic functions includes the parent function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] along with all of its transformations: shifts, stretches, compressions, and reflections.

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

Using the inputs and outputs from the table above, we can build another table to observe the relationship between points on the graphs of the inverse functions [latex]f\left(x\right)={2}^{x}[/latex] and [latex]g\left(x\right)={\mathrm{log}}_{2}\left(x\right)[/latex].

As we would expect, the x and y-coordinates are reversed for the inverse functions. The figure below shows the graphs of f and g.

Notice that the graphs of [latex]f\left(x\right)={2}^{x}[/latex] and [latex]g\left(x\right)={\mathrm{log}}_{2}\left(x\right)[/latex] are reflections about the line y = x since they are inverses of each other.

Observe the following from the graph:

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

The graphs below show how changing the base b in [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x\right)[/latex] can affect the graphs. Observe that the graphs compress vertically as the value of the base increases. (Note: recall that the function [latex]\mathrm{ln}\left(x\right)[/latex] is base [latex]e\approx \text{2}.\text{718.) and [latex]\mathrm{ln}\left(x\right)[/latex] is base 10}[/latex]

The graphs of three logarithmic functions with different bases all greater than 1.

Example: Graphing a Logarithmic Function Of the Form [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x\right)[/latex]

Graph [latex]f\left(x\right)={\mathrm{log}}_{5}\left(x\right)[/latex]. State the domain, range, and asymptote.

Show Solution

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

• Since b = 5 is greater than one, we know the function is increasing. The left tail of the graph will approach the vertical asymptote x = 0, and the right tail will increase slowly without bound.
• The x-intercept is [latex]\left(1,0\right)[/latex].
• The key point [latex]\left(5,1\right)[/latex] is on the graph.
• We draw and label the asymptote, plot and label the points, and draw a smooth curve through the points.

The domain is [latex]\left(0,\infty \right)[/latex], the range is [latex]\left(-\infty ,\infty \right)[/latex], and the vertical asymptote is x = 0.

Try It

Graph [latex]f\left(x\right)={\mathrm{log}}_{\frac{1}{5}}\left(x\right)[/latex]. State the domain, range, and asymptote.

Show Solution

The domain is [latex]\left(0,\infty \right)[/latex], the range is [latex]\left(-\infty ,\infty \right)[/latex], and the vertical asymptote is x = 0.

Horizontal and Vertical Shifts of Logarithmic Functions

As we mentioned in the beginning of the section, transformations of logarithmic functions behave similar to those of other parent functions. We can shift, stretch, compress, and reflect the parent function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] without loss of shape.

Graphing a Horizontal Shift of [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x\right)[/latex]

When a constant c is added to the input of the parent function [latex]f\left(x\right)=\text{log}_{b}\left(x\right)[/latex], 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 [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x\right)[/latex] alongside the shift left, [latex]g\left(x\right)={\mathrm{log}}_{b}\left(x+c\right)[/latex], and the shift right, [latex]h\left(x\right)={\mathrm{log}}_{b}\left(x-c\right)[/latex] where c > 0.The graphs below summarize the changes in the x-intercepts, vertical asymptotes, and equations of a logarithmic function that has been shifted either right or left.

Example: Graphing a Horizontal Shift of the Parent Function [latex]y=\text{log}_{b}\left(x\right)[/latex]

Sketch the horizontal shift [latex]f\left(x\right)={\mathrm{log}}_{3}\left(x - 2\right)[/latex] alongside its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.

Show Solution

Since the function is [latex]f\left(x\right)={\mathrm{log}}_{3}\left(x - 2\right)[/latex], we notice [latex]x+\left(-2\right)=x - 2[/latex].

Thus c = –2, so c < 0. This means we will shift the function [latex]f\left(x\right)={\mathrm{log}}_{3}\left(x\right)[/latex] right 2 units.

The vertical asymptote is [latex]x=-\left(-2\right)[/latex] or x = 2.

Consider the three key points from the parent function: [latex]\left(\frac{1}{3},-1\right)[/latex], [latex]\left(1,0\right)[/latex], and [latex]\left(3,1\right)[/latex].

The new coordinates are found by adding 2 to the x coordinates of each point.

Plot and label the points [latex]\left(\frac{7}{3},-1\right)[/latex], [latex]\left(3,0\right)[/latex], and [latex]\left(5,1\right)[/latex].

The domain is [latex]\left(2,\infty \right)[/latex], the range is [latex]\left(-\infty ,\infty \right)[/latex], and the vertical asymptote is x = 2.

Try It

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

Show Solution

The domain is [latex]\left(-4,\infty \right)[/latex], the range [latex]\left(-\infty ,\infty \right)[/latex], and the asymptote x = –4.

Graphing a Vertical Shift of [latex]y=\text{log}_{b}\left(x\right)[/latex]

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

Example: Graphing a Vertical Shift of the Parent Function [latex]y=\text{log}_{b}\left(x\right)[/latex]

Sketch a graph of [latex]f\left(x\right)={\mathrm{log}}_{3}\left(x\right)-2[/latex] alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.

Show Solution

Since the function is [latex]f\left(x\right)={\mathrm{log}}_{3}\left(x\right)-2[/latex], we notice d = –2. Thus d < 0.

This means we will shift the function [latex]f\left(x\right)={\mathrm{log}}_{3}\left(x\right)[/latex] down 2 units.

The vertical asymptote is x = 0.

Consider the three key points from the parent function, [latex]\left(\frac{1}{3},-1\right)[/latex], [latex]\left(1,0\right)[/latex], and [latex]\left(3,1\right)[/latex].

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

Label the points [latex]\left(\frac{1}{3},-3\right)[/latex], [latex]\left(1,-2\right)[/latex], and [latex]\left(3,-1\right)[/latex].

The domain is [latex]\left(0,\infty \right)[/latex], the range is [latex]\left(-\infty ,\infty \right)[/latex], and the vertical asymptote is x = 0.

Try It

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

Show Solution

The domain is [latex]\left(0,\infty \right)[/latex], the range is [latex]\left(-\infty ,\infty \right)[/latex], and the vertical asymptote is x = 0.

Stretching, Compressing, or Reflecting a Logarithmic Function

Graphing Stretches and Compressions of [latex]y=\text{log}_{b}\left(x\right)[/latex]

When the parent function [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x\right)[/latex] 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 [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x\right)[/latex] alongside the vertical stretch, [latex]g\left(x\right)=a{\mathrm{log}}_{b}\left(x\right)[/latex], and the vertical compression, [latex]h\left(x\right)=\frac{1}{a}{\mathrm{log}}_{b}\left(x\right)[/latex].The graphs below summarize the key features of the resulting graphs of vertical stretches and compressions of logarithmic functions.

Example: Graphing a Stretch or Compression of the Parent Function [latex]y=\text{log}_{b}\left(x\right)[/latex]

Sketch the graph of [latex]f\left(x\right)=2{\mathrm{log}}_{4}\left(x\right)[/latex] alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.

Show Solution

Since the function is [latex]f\left(x\right)=2{\mathrm{log}}_{4}\left(x\right)[/latex], we will note that = 2.

This means we will stretch the function [latex]f\left(x\right)={\mathrm{log}}_{4}\left(x\right)[/latex] by a factor of 2.

The vertical asymptote is x = 0.

Consider the three key points from the parent function, [latex]\left(\frac{1}{4},-1\right)[/latex], [latex]\left(1,0\right)[/latex], and [latex]\left(4,1\right)[/latex].

The new coordinates are found by multiplying the y coordinates of each point by 2.

Label the points [latex]\left(\frac{1}{4},-2\right)[/latex], [latex]\left(1,0\right)[/latex], and [latex]\left(4,\text{2}\right)[/latex].

The domain is [latex]\left(0,\infty \right)[/latex], the range is [latex]\left(-\infty ,\infty \right)[/latex], and the vertical asymptote is x = 0.

Try It

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

Show Solution

The domain is [latex]\left(0,\infty \right)[/latex], the range is [latex]\left(-\infty ,\infty \right)[/latex], and the vertical asymptote is x = 0.

Example: Combining a Shift and a Stretch

Sketch the graph of [latex]f\left(x\right)=5\mathrm{log}\left(x+2\right)[/latex]. State the domain, range, and asymptote.

Show Solution

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

The domain is [latex]\left(-2,\infty \right)[/latex], the range is [latex]\left(-\infty ,\infty \right)[/latex], and the vertical asymptote is x = –2.

Try It

Sketch a graph of the function [latex]f\left(x\right)=3\mathrm{log}\left(x - 2\right)+1[/latex]. State the domain, range, and asymptote.

Show Solution

The domain is [latex]\left(2,\infty \right)[/latex], the range is [latex]\left(-\infty ,\infty \right)[/latex], and the vertical asymptote is x = 2.

Graphing Reflections of [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x\right)[/latex]

When the parent function [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x\right)[/latex] 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 [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x\right)[/latex] alongside the reflection about the x-axis, [latex]g\left(x\right)={\mathrm{-log}}_{b}\left(x\right)[/latex], and the reflection about the y-axis, [latex]h\left(x\right)={\mathrm{log}}_{b}\left(-x\right)[/latex].

The graphs below summarize the key characteristics of reflecting [latex]f(x) = \log_{b}{x}[/latex] horizontally and vertically.

Example: Graphing a Reflection of a Logarithmic Function

Sketch a graph of [latex]f\left(x\right)=\mathrm{log}\left(-x\right)[/latex] alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.

Show Solution

Before graphing [latex]f\left(x\right)=\mathrm{log}\left(-x\right)[/latex], identify the behavior and key points for the graph.

• Since b = 10 is greater than one, we know that the parent function is increasing. Since the input value is multiplied by –1, f is a reflection of the parent graph about the y-axis. Thus, [latex]f\left(x\right)=\mathrm{log}\left(-x\right)[/latex] will be decreasing as x moves from negative infinity to zero, and the right tail of the graph will approach the vertical asymptote x = 0.
• The x-intercept is [latex]\left(-1,0\right)[/latex].
• We draw and label the asymptote, plot and label the points, and draw a smooth curve through the points.

The domain is [latex]\left(-\infty ,0\right)[/latex], the range is [latex]\left(-\infty ,\infty \right)[/latex], and the vertical asymptote is x = 0.

Try It

Graph [latex]f\left(x\right)=-\mathrm{log}\left(-x\right)[/latex]. State the domain, range, and asymptote.

Show Solution

The domain is [latex]\left(-\infty ,0\right)[/latex], the range is [latex]\left(-\infty ,\infty \right)[/latex], and the vertical asymptote is x = 0.

Summarizing Transformations of Logarithmic Functions

Now that we have worked with each type of transformation for the logarithmic function, we can summarize each in the table below to arrive at the general equation for transforming exponential functions.

Example: Finding the Equation from a Graph

Find a possible equation for the common logarithmic function graphed below.

Show Solution

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

[latex]f\left(x\right)=-a\mathrm{log}\left(x+2\right)+k[/latex]

It appears the graph passes through the points [latex]\left(-1,1\right)[/latex] and [latex]\left(2,-1\right)[/latex]. Substituting [latex]\left(-1,1\right)[/latex],

[latex]\begin{array}{llll}1=-a\mathrm{log}\left(-1+2\right)+k\hfill & \text{Substitute }\left(-1,1\right).\hfill \\ 1=-a\mathrm{log}\left(1\right)+k\hfill & \text{Arithmetic}.\hfill \\ 1=k\hfill & \text{log(1)}=0.\hfill \end{array}[/latex]

Next, substituting [latex]\left(2,-1\right)[/latex],

[latex]\begin{array}{llllll}-1=-a\mathrm{log}\left(2+2\right)+1\hfill & \hfill & \text{Plug in }\left(2,-1\right).\hfill \\ -2=-a\mathrm{log}\left(4\right)\hfill & \hfill & \text{Arithmetic}.\hfill \\ \text{ }a=\frac{2}{\mathrm{log}\left(4\right)}\hfill & \hfill & \text{Solve for }a.\hfill \end{array}[/latex]

This gives us the equation [latex]f\left(x\right)=-\frac{2}{\mathrm{log}\left(4\right)}\mathrm{log}\left(x+2\right)+1[/latex].

Analysis of the Solution

We can verify this answer by comparing the function values in the table below with the points on the graph in this example.

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 below.

Solution

[latex]f\left(x\right)=2\mathrm{ln}\left(x+3\right)-1[/latex]

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.
• The graph of the parent function [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x\right)[/latex] has an x-intercept at [latex]\left(1,0\right)[/latex], domain [latex]\left(0,\infty \right)[/latex], range [latex]\left(-\infty ,\infty \right)[/latex], vertical asymptote x = 0, and
  • if b > 1, the function is increasing.
  • if 0 < b < 1, the function is decreasing.
• The equation [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x+c\right)[/latex] shifts the parent function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] horizontally
  • left c units if c > 0.
  • right c units if c < 0.
• The equation [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x\right)+d[/latex] shifts the parent function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] vertically
  • up d units if d > 0.
  • down d units if d < 0.
• For any constant a > 0, the equation [latex]f\left(x\right)=a{\mathrm{log}}_{b}\left(x\right)[/latex]
  • stretches the parent function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] vertically by a factor of a if |a| > 1.
  • compresses the parent function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] vertically by a factor of a if |a| < 1.
• When the parent function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] 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.
  • The equation [latex]f\left(x\right)=-{\mathrm{log}}_{b}\left(x\right)[/latex] represents a reflection of the parent function about the x-axis.
  • The equation [latex]f\left(x\right)={\mathrm{log}}_{b}\left(-x\right)[/latex] represents a reflection of the parent function about the y-axis.
  • A graphing calculator may be used to approximate solutions to some logarithmic equations.
• All transformations of the logarithmic function can be summarized by the general equation [latex] f\left(x\right)=a{\mathrm{log}}_{b}\left(x+c\right)+d[/latex].
• Given an equation with the general form [latex] f\left(x\right)=a{\mathrm{log}}_{b}\left(x+c\right)+d[/latex], we can identify the vertical asymptote x = –c for the transformation.
• Using the general equation [latex]f\left(x\right)=a{\mathrm{log}}_{b}\left(x+c\right)+d[/latex], we can write the equation of a logarithmic function given its graph.