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.