Graphs of Logarithmic Functions

Characteristics of Graphs of Logarithmic Functions

Before working with graphs, we will take a look at the domain (the set of input values) for which the logarithmic function is defined.

Recall that the exponential function is defined as $y={b}^{x}$ for any real number x and constant $b>0$ , $b\ne 1$ , where

The domain of y is $\left(-\infty ,\infty \right)$ .
The range of y is $\left(0,\infty \right)$ .

In the last section we learned that the logarithmic function $y={\mathrm{log}}_{b}\left(x\right)$ is the inverse of the exponential function $y={b}^{x}$ . So, as inverse functions:

The domain of $y={\mathrm{log}}_{b}\left(x\right)$ is the range of $y={b}^{x}$ : $\left(0,\infty \right)$ .
The range of $y={\mathrm{log}}_{b}\left(x\right)$ is the domain of $y={b}^{x}$ : $\left(-\infty ,\infty \right)$ .

Graphing a Logarithmic Function Using a Table of Values

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. We begin with the function $y={\mathrm{log}}_{b}\left(x\right)$ . Because every logarithmic function of this form is the inverse of an exponential function of 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 logarithmic form $x={\mathrm{log}}_{2}\left(y\right)$ in the table below.

x	–3	–2	–1	0	1	2	3
${2}^{x}=y$	$\frac{1}{8}$	$\frac{1}{4}$	$\frac{1}{2}$	1	2	4	8
${\mathrm{log}}_{2}\left(y\right)=x$	–3	–2	–1	0	1	2	3

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 $g\left(x\right)={\mathrm{log}}_{2}\left(x\right)$ .

$f\left(x\right)={2}^{x}$	$\left(-3,\frac{1}{8}\right)$	$\left(-2,\frac{1}{4}\right)$	$\left(-1,\frac{1}{2}\right)$	$\left(0,1\right)$	$\left(1,2\right)$	$\left(2,4\right)$	$\left(3,8\right)$
$g\left(x\right)={\mathrm{log}}_{2}\left(x\right)$	$\left(\frac{1}{8},-3\right)$	$\left(\frac{1}{4},-2\right)$	$\left(\frac{1}{2},-1\right)$	$\left(1,0\right)$	$\left(2,1\right)$	$\left(4,2\right)$	$\left(8,3\right)$

As we would expect, the x and y-coordinates are reversed for the inverse functions. The figure below shows the graphs of f and g.

Graph of two functions, f(x)=2^x and g(x)=log_2(x), with the line y=x denoting the axis of symmetry.

Notice that the graphs of $f\left(x\right)={2}^{x}$ and $g\left(x\right)={\mathrm{log}}_{2}\left(x\right)$ are reflections about the line y = x since they are inverses of each other.

Observe the following from the graph:

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

Watch the following video for an excellent demonstration on how to graph a logarithmic function…

A General Note: Characteristics of the Graph of the Function $f\left(x\right)={\mathrm{log}}_{b}\left(x\right)$

For any real number x and constant b > 0, $b\ne 1$ , we can see the following characteristics in the graph of $f\left(x\right)={\mathrm{log}}_{b}\left(x\right)$ :

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

How To: Given a logarithmic function Of the form $f\left(x\right)={\mathrm{log}}_{b}\left(x\right)$ , graph the function

If the function is in function notation, replace $f\left(x\right)$ with y.
Change the logarithmic equation to exponential form using the definition of the logarithm.
Make a table of points, choosing the y-values first and finding the x-values, making sure to include the x-intercept.
Plot the points. Make sure to include the x-intercept and the key point.
Draw a smooth curve through the points.
Draw and label the vertical asymptote.

Example: Graphing a Logarithmic Function Of the Form $f\left(x\right)={\mathrm{log}}_{b}\left(x\right)$

Graph $f\left(x\right)={\mathrm{log}}_{5}\left(x\right)$ . State the domain, range, and asymptote.

Show 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 $\left(1,0\right)$ .
The key point $\left(5,1\right)$ is on the graph.
We draw and label the asymptote, plot and label the points, and draw a smooth curve through the points.

Graph of f(x)=log_5(x) with labeled points at (1, 0) and (5, 1). The y-axis is the asymptote.

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

Try It

Graph $f\left(x\right)={\mathrm{log}}_{\frac{1}{5}}\left(x\right)$ . State the domain, range, and asymptote.

Show Solution

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

Graph of f(x)=log_(1/5)(x) with labeled points at (1/5, 1) and (1, 0). The y-axis is the asymptote.

The graphs below show how changing the base b in $f\left(x\right)={\mathrm{log}}_{b}\left(x\right)$ can affect the graphs. Observe that the graphs compress vertically as the value of the base increases. (Note: recall that the function $\mathrm{ln}\left(x\right)$ is base $e\approx \text{2}.\text{718}$ and $\mathrm{log} (x)$ is base 10.

Graph of three equations: y=log_2(x) in blue, y=ln(x) in orange, and y=log(x) in red. The y-axis is the asymptote.

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

Key Concepts

The inverse of an exponential function is a logarithmic function, and the inverse of a logarithmic function is an exponential function.
Logarithmic equations can be written in an equivalent exponential form using the definition of a logarithm.
Exponential equations can be written in an equivalent logarithmic form using the definition of a logarithm.
Logarithmic functions with base b can be evaluated mentally using previous knowledge of powers of b.
Common logarithms can be evaluated mentally using previous knowledge of powers of 10.
When common logarithms cannot be evaluated mentally, a calculator can be used.
Natural logarithms can be evaluated using a calculator.
To find the domain of a logarithmic function, set up an inequality showing the argument greater than zero and solve for x.
The graph of the parent function $f (x) = {l o g}_{b} (x)$ has an x-intercept at $(1, 0)$ , domain $(0, \infty)$ , range $(- \infty, \infty)$ , vertical asymptote x = 0, and
- if b > 1, the function is increasing.
- if 0 < b < 1, the function is decreasing.

Glossary

common logarithm: the exponent to which 10 must be raised to get x; ${\mathrm{log}}_{10}\left(x\right)$ is written simply as $\mathrm{log}\left(x\right)$

logarithm: the exponent to which b must be raised to get x; written $y={\mathrm{log}}_{b}\left(x\right)$

natural logarithm: the exponent to which the number e must be raised to get x; ${\mathrm{log}}_{e}\left(x\right)$ is written as $\mathrm{ln}\left(x\right)$

Unit 4: Exponential & Logarithmic Functions & Models