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.

  1. Draw and label the vertical asymptote, x=0.
  2. Plot the x-intercept, (1,0).
  3. Plot the key point (b,1).
  4. Draw a smooth curve through the points.
  5. 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.