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 [latex]y={b}^{x}[/latex] for any real number x and constant [latex]b>0[/latex], [latex]b\ne 1[/latex], where

  • The domain of y is [latex]\left(-\infty ,\infty \right)[/latex].
  • The range of y is [latex]\left(0,\infty \right)[/latex].

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

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

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 [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex]. Because every logarithmic function of this form is the inverse of an exponential function of 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 logarithmic form [latex]x={\mathrm{log}}_{2}\left(y\right)[/latex] in the table below.

x –3 –2 –1 0 1 2 3
[latex]{2}^{x}=y[/latex] [latex]\frac{1}{8}[/latex] [latex]\frac{1}{4}[/latex] [latex]\frac{1}{2}[/latex] 1 2 4 8
[latex]{\mathrm{log}}_{2}\left(y\right)=x[/latex] –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 [latex]f\left(x\right)={2}^{x}[/latex] and [latex]g\left(x\right)={\mathrm{log}}_{2}\left(x\right)[/latex].

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

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 [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 since they are inverses of each other.

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].

 

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 [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x\right)[/latex]

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

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

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

Example: Graphing a Logarithmic Function Of 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.

Try It

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


The graphs below show 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] is base [latex]e\approx \text{2}.\text{718}[/latex] and [latex]\mathrm{log} (x)[/latex] 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 [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x\right)[/latex] has an x-intercept at [latex]\left(1,0\right)[/latex], domain [latex]\left(0,\infty \right)[/latex], range [latex]\left(-\infty ,\infty \right)[/latex], 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; [latex]{\mathrm{log}}_{10}\left(x\right)[/latex] is written simply as [latex]\mathrm{log}\left(x\right)[/latex]
logarithm
the exponent to which b must be raised to get x; written [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex]
natural logarithm
the exponent to which the number e must be raised to get x; [latex]{\mathrm{log}}_{e}\left(x\right)[/latex] is written as [latex]\mathrm{ln}\left(x\right)[/latex]