158 Graph 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 [latex]y={\mathrm{log}}_{b}\left(x\right)\\[/latex] along with all 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 with 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 [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’d expect, the x– and y-coordinates are reversed for the inverse functions. The figure below shows the graph of f and g.

Figure 2. 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.

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].

A General Note: Characteristics of the Graph of the Parent Function, f(x) = log_b(x)

For any real number x and constant b > 0, [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: x = 0
• domain: [latex]\left(0,\infty \right)\\[/latex]
• range: [latex]\left(-\infty ,\infty \right)\\[/latex]
• x-intercept: [latex]\left(1,0\right)\\[/latex] and key point [latex]\left(b,1\right)\\[/latex]
• y-intercept: none
• increasing if [latex]b>1\\[/latex]
• decreasing if 0 < b < 1

Figure 3 shows 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] has base [latex]e\approx \text{2}.\text{718.)}\\[/latex]

Example 3: Graphing a Logarithmic Function with 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.

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.

Figure 5. 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.