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 [latex]f\left(x\right)={b}^{x}\text{ }[/latex] for any real number [latex]x[/latex] and constant [latex]b>0,[/latex] [latex]b\ne 1,[/latex] where

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

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

• The domain of [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x\right)[/latex] is the range of [latex]f^{-1}\left(x\right)={b}^{x}:[/latex] [latex]\left(0,\infty \right).[/latex]
• The range of [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x\right)[/latex] is the domain of [latex]f^{-1}\left(x\right)={b}^{x}:[/latex] [latex]\left(-\infty ,\infty \right).[/latex]

Transformations of the logarithmic function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] behave similarly to those of other functions. Just as with toolkit and exponential functions, we can apply the four types of transformations—shifts, stretches, compressions, and reflections—to the original function without loss of shape.

In Section 2.3, Graphs of Exponential Functions we saw that certain transformations can change the range of [latex]y={b}^{x}.[/latex] Similarly, applying transformations to the function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] can change the domain. When finding the domain of a logarithmic function, therefore, it is important to remember that we can only take the logarithm of positive real numbers. That is, the value of the input expression of the logarithmic function must be greater than zero.

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

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

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 function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] along with all its transformations: shifts, stretches, compressions, and reflections.

We begin with the function [latex]y={\mathrm{log}}_{b}\left(x\right).[/latex] Because every logarithmic function of this form is the inverse of an exponential function with base b, 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 [latex]x={\mathrm{log}}_{2}\left(y\right)[/latex] in Table 1.

Using the inputs and outputs from Table 1, 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] See Table 2.

As we’d expect, the x– and y-coordinates are reversed for the inverse functions. Figure 2 shows the graph of [latex]f[/latex] and [latex]g.[/latex]

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

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]

Recall that for exponential growth functions like [latex]f\left(x\right)=2^x[/latex], we observed that as x decreased without bound, [latex]f\left(x\right)[/latex] gets closer and closer to zero from above. We concluded that exponential growth functions of the form [latex]f(x)=ab^x[/latex] have a horizontal asymptote of [latex]y=0[/latex] on one side.  We created tables supporting this idea numerically.  Using the inverse relationship, we will study what happens in the logarithmic function as the input gets closer and closer to zero from the right of zero or through values of x slightly larger than zero.

To create a table of values to explore this idea, we choose input values that get closer and closer to zero from the right side and then evaluate [latex]g\left(x\right)={\mathrm{log}}_{2}\left(x\right).[/latex]  We choose values slightly larger than zero because the logarithm is defined only for positive values and we want to observe what happens near the boundary of the domain.  See Table 3.

As [latex]x[/latex] gets closer and closer to zero from the right (or from the positive side), the function values decrease without bound (go toward minus infinity).  Referring back to Figure 2,  we observe that the graph of [latex]g\left(x\right)={\mathrm{log}}_{2}\left(x\right)[/latex]  decreases without bound or decreases rapidly as [latex]x[/latex] approaches zero from the right.

We use arrow notation to express these ideas. See Table 4.

Note that when technology is used to graph logarithmic functions, it often appears that there is a vertical intercept rather than a vertical asymptote. Creating a table of values demonstrates that, in fact, there is a vertical asymptote. 

Characteristics of the Graph of the Function, f(x) = log_b(x)

For any real number [latex]x[/latex] and constant [latex]b>0,[/latex] [latex]b\ne 1,[/latex] we can see the following characteristics in the graph of [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x\right):[/latex]

• one-to-one function
• vertical asymptote: [latex]x=0[/latex]
• domain: [latex]\left(0,\infty \right)[/latex]
• range: [latex]\left(-\infty ,\infty \right)[/latex]
• horizontal intercept: [latex]\left(1,0\right)[/latex] and key point [latex]\left(b,1\right)[/latex]
• vertical intercept: none
• increasing if [latex]b>1[/latex] and decreasing if [latex]0<b<1[/latex]

See Figure 3.

Figure 3.

For [latex]b > 1,[/latex] the base of the logarithm effects how quickly the graph increases as [latex]x[/latex] gets larger or decreases as [latex]x[/latex] goes to zero. Figure 5 shows the graphs of three logarithmic functions [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x\right)[/latex] with the base [latex]b=2,[/latex] [latex]b=e\approx 2.718,[/latex] and [latex]b=10[/latex]. Observe that the graphs compress vertically as the value of the base increases. The key points for the graphs are (2, 1), (e, 1) and (10, 1) respectively showing that base [latex]b=2[/latex] reaches an output value of 1 more quickly than base e or 10.

Figure 5. The graphs of three logarithmic functions with different bases, all greater than 1.

Graphing Transformations of Logarithmic Functions

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

Graphing a Horizontal Shift of y = log_b(x)

We will begin by looking at a horizontal shift of the function [latex]y={\mathrm{log}}_{b}\left(x\right).[/latex] Consider the function [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x-c\right),[/latex] where [latex]c[/latex] is a constant. If [latex]c[/latex] is positive, then the horizontal shift is to the right and if [latex]c[/latex] is negative, the horizontal shift is to the left. Note that, as we observed in earlier sections, because the general formula for the horizontal shift contains a minus sign, [latex]c[/latex] will have the opposite sign of what you observe in the formula.

Figure 6

Horizontal Shifts of the Function y = log_b(x)

For any constant [latex]c,[/latex] the function [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x-c\right)[/latex]

• shifts the function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] right [latex]c[/latex] units if [latex]c>0.[/latex]
• shifts the function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] left [latex]c[/latex] units if [latex]c<0.[/latex]
• has the vertical asymptote [latex]x=c.[/latex]
• has domain [latex]\left(c,\infty \right).[/latex]
• has range [latex]\left(-\infty ,\infty \right).[/latex]

How To

Given a logarithmic function with the form [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x-c\right),[/latex] graph the translation.

1. Determine the value for [latex]c.[/latex]  It will have the opposite sign of what you see in the simplified formula.
2. Identify the horizontal shift:
   • If [latex]c>0,[/latex] shift the graph of [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] right [latex]c[/latex] units.
   • If [latex]c<0,[/latex] shift the graph of [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] left [latex]c[/latex] units.
3. Draw the vertical asymptote [latex]x=c.[/latex]
4. Identify two or three key points from the function [latex]y={\mathrm{log}}_{b}\left(x\right).[/latex] Find new coordinates for the shifted functions by adding [latex]c[/latex] to the [latex]x[/latex] coordinate.
5. Label the points.
6. The domain is [latex]\left(c,\infty \right),[/latex] the range is [latex]\left(-\infty ,\infty \right),[/latex] and the vertical asymptote is [latex]x=c.[/latex]

example 4:  Graphing a Horizontal Shift of the Function y = log_b(x)

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

Show Solution

Since the function is [latex]f\left(x\right)={\mathrm{log}}_{3}\left(x-2\right),[/latex] the input expression is [latex]x-\left(2\right).[/latex]  Thus [latex]c=2,[/latex] so [latex]c>0.[/latex] This means we will shift the function [latex]y={\mathrm{log}}_{3}\left(x\right)[/latex] right 2 units.

The vertical asymptote is [latex]x=2.[/latex]

Consider the two key points from the function [latex]y={\mathrm{log}}_{3}\left(x\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 [latex]x[/latex] coordinates.

Label the points [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 [latex]x=2.[/latex]

Figure 7.

Try it #4

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

Show Solution

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

Graphing a Vertical Shift of y = log_b(x)

When a constant [latex]d[/latex] is added to the function [latex]y={\mathrm{log}}_{b}\left(x\right),[/latex] the result is a vertical shift [latex]d[/latex] units in the direction of the sign of [latex]d.[/latex] To visualize vertical shifts, we can observe the general graph of the function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] alongside the shift, [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x\right)+d.[/latex] See Figure 8.

Figure 8

Vertical Shifts of the Function y = log_b(x)

For any constant [latex]d,[/latex] the function [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x\right)+d[/latex]

• shifts the function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] up [latex]d[/latex] units if [latex]d>0.[/latex]
• shifts the function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] down [latex]d[/latex] units if [latex]d<0.[/latex]
• has the vertical asymptote [latex]x=0.[/latex]
• has domain [latex]\left(0,\infty \right).[/latex]
• has range [latex]\left(-\infty ,\infty \right).[/latex]

How To

Given a logarithmic function with the form [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x\right)+d,[/latex] graph the translation.

1. Identify the vertical shift:
   • If [latex]d>0,[/latex] shift the graph of [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] up [latex]d[/latex] units.
   • If [latex]d<0,[/latex] shift the graph of [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] down [latex]d[/latex] units.
2. Draw the vertical asymptote [latex]x=0.[/latex]
3. Identify two or three key points from the function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex]. Find new coordinates for the shifted functions by adding [latex]d[/latex] to the [latex]y[/latex] coordinate.
4. Label the points.
5. The domain is [latex]\left(0,\infty \right),[/latex] the range is [latex]\left(-\infty ,\infty \right),[/latex] and the vertical asymptote is [latex]x=0.[/latex]

example 5:  Graphing a Vertical Shift of the Function y = log_b(x)

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

Show Solution

Since the function is [latex]f\left(x\right)={\mathrm{log}}_{3}\left(x\right)-2,[/latex] we observe that  [latex]d=–2.[/latex] Thus [latex]d<0.[/latex] This means we will shift the function [latex]y={\mathrm{log}}_{3}\left(x\right)[/latex] down 2 units.

The vertical asymptote is [latex]x=0.[/latex]

Consider the three key points from the function [latex]y={\mathrm{log}}_{3}\left(x\right)[/latex], [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. 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 [latex]x=0.[/latex]

Figure 9.

Try it #5

Sketch a graph of [latex]f\left(x\right)={\mathrm{log}}_{2}\left(x\right)+2[/latex] alongside the function [latex]y={\mathrm{log}}_{2}\left(x\right)[/latex]. Include the key points and vertical asymptote on the graph. State the domain, range, and vertical 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 [latex]x=0.[/latex]

Graphing Vertical Stretches and Compressions of y = log_b(x)

When the function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] is multiplied by a constant [latex]a>0,[/latex] the result is a vertical stretch or compression of the original graph. To visualize stretches and compressions, we set [latex]a>0[/latex] and observe the general graph of the function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] alongside the vertical stretch or compression, [latex]f\left(x\right)=a{\mathrm{log}}_{b}\left(x\right).[/latex] See Figure 10.

Figure 10

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

For any constant [latex]a>1,[/latex] the function [latex]f\left(x\right)=a{\mathrm{log}}_{b}\left(x\right)[/latex]

• stretches the function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] vertically by a factor of [latex]a[/latex] if [latex]a>1.[/latex]
• compresses the  function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] vertically by a factor of [latex]a[/latex] if [latex]0<a<1.[/latex]
• has the vertical asymptote [latex]x=0.[/latex]
• has the horizontal intercept [latex]\left(1,0\right).[/latex]
• has domain [latex]\left(0,\infty \right).[/latex]
• has range [latex]\left(-\infty ,\infty \right).[/latex]

How To

Given a logarithmic function with the form  [latex]f\left(x\right)=a{\mathrm{log}}_{b}\left(x\right),[/latex] [latex]a>0,[/latex] graph the translation.

1. Identify the vertical stretch or compressions:
   • If [latex]|a|>1,[/latex] the graph of [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] is stretched by a factor of [latex]a[/latex] units.
   • If [latex]|a|<1,[/latex] the graph of [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] is compressed by a factor of [latex]a[/latex] units.
2. Draw the vertical asymptote [latex]x=0.[/latex]
3. Identify two or three key points from the function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex]. Find new coordinates for the stretched or compressed function by multiplying the [latex]y[/latex] coordinates by [latex]a.[/latex]
4. Label the points.
5. The domain is [latex]\left(0,\infty \right),[/latex] the range is [latex]\left(-\infty ,\infty \right),[/latex] and the vertical asymptote is [latex]x=0.[/latex]

example 6:  Graphing a Stretch or Compression of the Function y = log_b(x)

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

Show Solution

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

The vertical asymptote is [latex]x=0.[/latex]

Consider the two key points from the function [latex]y={\mathrm{log}}_{4}\left(x\right)[/latex],  [latex]\left(1,0\right),[/latex] and [latex]\left(4,1\right).[/latex] The new coordinates are found by multiplying the [latex]y[/latex] coordinates by 2. Label the points [latex]\left(1,0\right),[/latex] and [latex]\left(4,\text{2}\right).[/latex]

The domain is [latex]\left(0,\text{ }\infty \right),[/latex] the range is [latex]\left(-\infty ,\infty \right),[/latex] and the vertical asymptote is [latex]x=0.[/latex]  See Figure 11.

Figure 11.

Try it #6

Sketch a graph of [latex]f\left(x\right)=\frac{1}{2}\text{ }{\mathrm{log}}_{4}\left(x\right)[/latex] alongside the function [latex]y={\mathrm{log}}_{4}\left(x\right)[/latex]. Include the key points and the vertical asymptote on the graph. State the domain, range, and vertical 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 [latex]x=0.[/latex]

example 7:  Combining a Shift and a Stretch

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

Show Solution

First, we move the graph of [latex]y=\mathrm{log}\left(x\right)[/latex] left 2 units, then stretch the function vertically by a factor of 5, as in Figure 12. The vertical asymptote will be shifted to [latex]x=-2.[/latex] 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 [latex]x=8[/latex] as the x-coordinate of one point to graph because when [latex]x=8,[/latex] [latex]x+2=10,[/latex] the base of the common logarithm.

Figure 12.

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

Try it #7

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 vertical asymptote.  Identify at least 2 key points on [latex]f\left(x\right)=3\mathrm{log}\left(x-2\right)+1.[/latex]

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 [latex]x=2.[/latex] Key points may include [latex]\left(3,\text{ }1\right),[/latex] and [latex]\left(12,\text{ }4\right).[/latex]

Graphing Reflections of y = log_b(x)

When the function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] is multiplied by [latex]-1[/latex] we are negating the output so, the result is a reflection about the x-axis. When the input is multiplied by [latex]-1,[/latex] the result is a reflection about the y-axis. To visualize reflections, we restrict [latex]b>1,[/latex] and observe the general graph of the function [latex]y={\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]

Figure 13

Reflections of the Function y = log_b(x)

The function [latex]f\left(x\right)={\mathrm{-log}}_{b}\left(x\right)[/latex]

• reflects the function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] about the x-axis, and
• has domain, [latex]\left(0,\infty \right),[/latex] range, [latex]\left(-\infty ,\infty \right),[/latex] and vertical asymptote, [latex]x=0,[/latex] which are unchanged from the original function.

The function [latex]f\left(x\right)={\mathrm{log}}_{b}\left(-x\right)[/latex]

• reflects the function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] about the y-axis,
• has domain [latex]\left(-\infty ,0\right),[/latex] and
• has range, [latex]\left(-\infty ,\infty \right),[/latex] and vertical asymptote, [latex]x=0,[/latex] which are unchanged from the original function.

How To

Given a logarithmic function with the  function [latex]y={\mathrm{log}}_{b}\left(x\right),[/latex] graph a translation.

If [latex]f\left(x\right)=-{\mathrm{log}}_{b}\left(x\right)[/latex]	If [latex]f\left(x\right)={\mathrm{log}}_{b}\left(-x\right)[/latex]
Draw the vertical asymptote, [latex]x=0.[/latex]	Draw the vertical asymptote, [latex]x=0.[/latex]
Plot the x-intercept, [latex]\left(1,0\right).[/latex]	Plot the x-intercept, [latex]\left(-1,0\right).[/latex]
Reflect the graph of the function [latex]fy={\mathrm{log}}_{b}\left(x\right)[/latex] about the x-axis. Key point [latex]\left(b,\text{ }1\right)[/latex] reflects to [latex]\left(b,\text{ }-1\right).[/latex]	Reflect the graph of the function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] about the y-axis. Key point [latex]\left(b,\text{ }1\right)[/latex] reflects to [latex]\left(-b,\text{ }1\right).[/latex]
Draw a smooth curve through the points.	Draw a smooth curve through the points.
State the domain, [latex]\left(0,\infty \right),[/latex] the range, [latex]\left(-\infty ,\infty \right),[/latex] and the vertical asymptote [latex]x=0.[/latex]	State the domain, [latex]\left(-\infty ,0\right),[/latex] the range, [latex]\left(-\infty ,\infty \right),[/latex] and the vertical asymptote [latex]x=0.[/latex]

example 8:  Graphing a Reflection of a Logarithmic Function

Sketch a graph of [latex]f\left(x\right)=\mathrm{log}\left(-x\right)[/latex] alongside the function [latex]y=\mathrm{log}\left(x\right)[/latex]. Include the key points and vertical asymptote on the graph. State the domain, range, and vertical 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 [latex]b=10[/latex] is greater than one, we know that the function [latex]y=\mathrm{log}\left(x\right)[/latex] is increasing. Since the input value is multiplied by [latex]-1,[/latex] [latex]f[/latex] is a reflection of the graph of [latex]y=\mathrm{log}\left(x\right)[/latex] about the y-axis. Thus, [latex]f\left(x\right)=\mathrm{log}\left(-x\right)[/latex] will be decreasing as [latex]x[/latex] moves from negative infinity to zero, and as x approaches zero from the left or through negative values, the graph will decrease without bound.
• The x-intercept is [latex]\left(-1,0\right).[/latex]
• We draw and label the vertical asymptote, plot and label the points, and draw a smooth curve through the points.

Figure 14

 

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

Try it #8

Graph [latex]f\left(x\right)=-\mathrm{log}\left(-x\right).[/latex] State the domain, range, and vertical 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 [latex]x=0.[/latex]  Since there is both a horizontal and vertical reflection, both the x and y coordinates will be negated.  Key points [latex]\left(1,\text{ } 0\right)[/latex] and [latex]\left(10,\text{ } 1\right)[/latex] on [latex]f\left(x\right)=\mathrm{log}\left(x\right)[/latex] will transform to [latex]\left(-1,\text{ } 0\right)[/latex] and [latex]\left(-10,\text{ } -1\right)[/latex] for the translated function.

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 5 to arrive at the general equation for translating exponential functions.

Table 5

Translations of the Function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex]
Translation	Form
Shift Horizontally [latex]c[/latex] units to the left or right Vertically [latex]d[/latex] units up or down	[latex]f\left(x\right)={\mathrm{log}}_{b}\left(x-c\right)+d[/latex]
Stretch and Compress Vertical stretch if [latex]|a|>1[/latex] Vertical compression if [latex]|a|<1[/latex]	[latex]f\left(x\right)=a{\mathrm{log}}_{b}\left(x\right)[/latex]
Reflect about the x-axis	[latex]f\left(x\right)=-{\mathrm{log}}_{b}\left(x\right)[/latex]
Reflect about the y-axis	[latex]f\left(x\right)={\mathrm{log}}_{b}\left(-x\right)[/latex]
General equation for all translations	[latex]f\left(x\right)=a{\mathrm{log}}_{b}\left(x-c\right)+d[/latex]

example 9:  Finding the Vertical Asymptote of a Logarithm Graph

What is the vertical asymptote of [latex]f\left(x\right)=-2{\mathrm{log}}_{3}\left(x+4\right)+5?[/latex]

Show Solution

The vertical asymptote is [latex]x=-4.[/latex]

Analysis

The coefficient, the base, and the upward translation do not affect the vertical asymptote.  The shift of the curve 4 units to the left shifts the vertical asymptote to [latex]x=-4.[/latex]

Try it #9

What is the vertical asymptote of [latex]f\left(x\right)=3+\mathrm{ln}\left(x-1\right)?[/latex]

Show Solution

[latex]x=1[/latex]

example 10:  Finding the Equation from a Graph

Find a possible equation for the common logarithmic function graphed in Figure 15.

Figure 15.

Show Solution

This graph has a vertical asymptote at [latex]x=-2[/latex] 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 form:

[latex]f\left(x\right)=-a\mathrm{log}\left(x+2\right)+d.[/latex][latex]\\[/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{align*}1&=-a\mathrm{log}\left(-1+2\right)+d && \text{Substitute }\left(-1,1\right).\hfill \\ 1&=-a\mathrm{log}\left(1\right)+d\hfill && \text{Arithmetic}.\hfill \\ 1&=d\hfill && \text{log(1)}=0.\hfill \end{align*}[/latex][latex]\\[/latex]

Therefore, we now have [latex]f\left(x\right)=-a\mathrm{log}\left(x+2\right)+1.[/latex] Next, substituting into this equation the point [latex]\left(2,–1\right)[/latex] we have,

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

We can verify this answer by comparing the function values in Table 6 with the points on the graph in Figure 15.

Table 6

[latex]x[/latex]	−1	0	1	2	3
[latex]f\left(x\right)[/latex]	1	0	−0.58496	−1	−1.3219
[latex]x[/latex]	4	5	6	7	8
[latex]f\left(x\right)[/latex]	−1.5850	−1.8074	−2	−2.1699	−2.3219

Try it #10

Give the equation of the natural logarithm graphed in Figure 16.  Note that the points [latex]\left(-2, -1\right) [/latex] and [latex]\left(e-3, 1\right)[/latex] are on the graph.

Figure 16.

Show Solution

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

Q&A

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 16. The graph approaches [latex]x=-3[/latex] (or thereabouts) more and more closely, so [latex]x=-3[/latex] is, or is very close to, the vertical asymptote. It approaches from the right, so the domain is all points to the right, [latex]\left\{x\text{ }|\text{ }x>-3\right\}.[/latex] 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 as [latex]x\to -{3}^{+},f\left(x\right)\to -\infty[/latex] and as [latex]x\to \infty ,f\left(x\right)\to \infty .[/latex]