Logarithmic Function Graphs and Characteristics: Learn It 2

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. The family of logarithmic functions includes the parent function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] along with all of its transformations: shifts, stretches, compressions, and reflections.

Recall that if an invertible function [latex]f(x)[/latex] contains a point, [latex]\left(a, b\right)[/latex], then the inverse function [latex]f^{-1}\left(x\right)[/latex] must contain the point [latex]\left(b, a\right)[/latex].
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 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.
[latex]x[/latex] [latex]–3[/latex] [latex]–2[/latex] [latex]–1[/latex] [latex]0[/latex] [latex]1[/latex] [latex]2[/latex] [latex]3[/latex]
[latex]{2}^{x}=y[/latex] [latex]\frac{1}{8}[/latex] [latex]\frac{1}{4}[/latex] [latex]\frac{1}{2}[/latex] [latex]1[/latex] [latex]2[/latex] [latex]4[/latex] [latex]8[/latex]
[latex]{\mathrm{log}}_{2}\left(y\right)=x[/latex] [latex]–3[/latex] [latex]–2[/latex] [latex]–1[/latex] [latex]0[/latex] [latex]1[/latex] [latex]2[/latex] [latex]3[/latex]

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 [latex]x[/latex]and [latex]y[/latex]-coordinates are reversed for the inverse functions. The figure below shows the graphs of [latex]f[/latex] and [latex]g[/latex].

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].

characteristics of the graph of the parent function [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x\right)[/latex]

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

 

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. Draw and label the vertical asymptote, [latex]x = 0[/latex].
  2. Plot the [latex]x[/latex]–intercept, [latex]\left(1,0\right)[/latex].
  3. Plot the key point [latex]\left(b,1\right)[/latex].
  4. Draw a smooth curve through the points.
  5. State the domain, [latex]\left(0,\infty \right)[/latex], the range, [latex]\left(-\infty ,\infty \right)[/latex], and the vertical asymptote, [latex]x = 0[/latex].
Graph [latex]f\left(x\right)={\mathrm{log}}_{5}\left(x\right)[/latex]. State the domain, range, and asymptote.

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