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 $y={\log }_b\left(x\right)$ along with all its transformations: shifts, stretches, compressions, and reflections.

We begin with the parent function $y={\log }_b\left(x\right)$. 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 $x={\log }_2\left(y\right)$ 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 $f\left(x\right)=2^x$ and $g\left(x\right)={\log }_2\left(x\right)$.

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 $f\left(x\right)=2^x$ and $g\left(x\right)={\log }_2\left(x\right)$ are reflections about the line y = x.

Observe the following from the graph:

• $f\left(x\right)=2^x$ has a y-intercept at $(0,1)$ and $g\left(x\right)={\log }_2\left(x\right)$ has an x-intercept at $(1,0)$.
• The domain of $f\left(x\right)=2^x$, $(-\mathrm{\infty },\mathrm{\infty })$, is the same as the range of $g\left(x\right)={\log }_2\left(x\right)$.
• The range of $f\left(x\right)=2^x$, $(0,\mathrm{\infty })$, is the same as the domain of $g\left(x\right)={\log }_2\left(x\right)$.

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, $b\ne 1$, we can see the following characteristics in the graph of $f\left(x\right)={\log }_b\left(x\right)$:

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

Figure 3 shows how changing the base b in $f\left(x\right)={\log }_b\left(x\right)$ can affect the graphs. Observe that the graphs compress vertically as the value of the base increases. (Note: recall that the function $\ln \left(x\right)$ has base $e\approx \text{2}.\text{718.)}$

Example 3: Graphing a Logarithmic Function with the Form $f\left(x\right)={\log }_b\left(x\right)$.

Graph $f\left(x\right)={\log }_5\left(x\right)$. 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.

Figure 5. The domain is $(0,\mathrm{\infty })$, the range is $(-\mathrm{\infty },\mathrm{\infty })$, and the vertical asymptote is x = 0.