Characterisitics of Graphs of Logarithmic Functions

Now, we will define the domain and range of a logarithmic function given an equation or a graph. We will also construct graphs of logarithmic functions given tables and equations.

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 1.

$x$	$-3$	$-2$	$-1$	$0$	$1$	$2$	$3$
$2^x=y$	$\frac{1}{8}$	$\frac{1}{4}$	$\frac{1}{2}$	$1$	$2$	$4$	$8$
$log_2(y) = x$	$-3$	$-2$	$-1$	$0$	$1$	$2$	$3$

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 $f(x)=2^x$ and $g(x)=log_2(x)$ . See Table 2.

$f(x)=2^x$	$(−3, \frac{1}{8})$	$(−2, \frac{1}{4})$	$(−1, \frac{1}{2})$	$(0, 1)$	$(1, 2)$	$(2, 4)$	$(3, 8)$
$g(x)=log_2(x)$	$(\frac{1}{8}, -3)$	$(\frac{1}{4}, -2)$	$(\frac{1}{2}, -1)$	$(1, 0)$	$(2, 1)$	$(4, 2)$	$(8, 3)$

Table 2

Figure 2 Notice that the graphs of $f(x)=2^x$ and $g(x)=log_2(x)$ are reflections about the line $y=x$ .

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 $g(x)=log_2(x)$ .

CHARACTERISTICS OF THE GRAPH OF THE PARENT FUNCTION, $f(x)=log_b(x)$ :

For any real number $x$ and constant $b > 0, b \neq 1$ , we can see the following characteristics in the graph of $f(x)=log_b(x)$ :

one-to-one function
vertical asymptote: $x=0$
domain: $(0, \infty)$
range: $(−\infty, \infty)$
$x$ -intercept: $(1,0)$ and key point $(b,1)$
$y$ -intercept: none
increasing if $b > 1$
decreasing if $0 < b < 1$

See Figure 3.

Figure 3

Figure 4 shows how changing the base $b$ in $f(x)=log_b(x)$ can affect the graphs. Observe that the graphs compress vertically as the value of the base increases. (Note: recall that the function $\ln(x)$ has base $e \approx 2.718$ .)

Figure 4 The graphs of three logarithmic functions with different bases, all greater than 1.

HOW TO

Given a logarithmic function with the form $f(x)=log_b(x)$ , graph the function.

Draw and label the vertical asymptote, $x=0$ .
Plot the $x$ -intercept, $(1,0)$ .
Plot the key point $(b,1)$ .
Draw a smooth curve through the points.
State the domain, $(0, \infty)$ , the range, $(−\infty, \infty)$ , and the vertical asymptote, $x=0$ .

EXAMPLE 3

Graphing a Logarithmic Function with the $f(x) = log_b(x)$ .

Graph $f(x)=log_5(x)$ . State the domain, range, and asymptote.

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 $(1,0)$ .
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 5).

Figure 5

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

TRY IT #3

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