Logarithmic Function Graphs and Characteristics: Apply It 1

  • Identify the domain of a logarithmic function
  • Graph logarithmic functions

Transformations of Logarithmic Functions

Now that we have worked with each type of transformation for the logarithmic function, we can summarize each in the table below to arrive at the general equation for transforming exponential functions.

Transformations of the Parent Function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex]
Translation Form
Shift

  • Horizontally [latex]c[/latex] units to the left
  • Vertically [latex]d[/latex] units up
 [latex]y={\mathrm{log}}_{b}\left(x+c\right)+d[/latex]
Stretch and Compression

  • Stretch if [latex]|a|>1[/latex]
  • Compression if [latex]|a|<1[/latex]
 [latex]y=a{\mathrm{log}}_{b}\left(x\right)[/latex]
Reflection about the [latex]x[/latex]-axis [latex]y=-{\mathrm{log}}_{b}\left(x\right)[/latex]
Reflection about the [latex]y[/latex]-axis [latex]y={\mathrm{log}}_{b}\left(-x\right)[/latex]
General equation for all transformations [latex]y=a{\mathrm{log}}_{b}\left(x+c\right)+d[/latex]

transformations of logarithmic functions

All transformations of the parent logarithmic function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] have the form

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

where the parent function, [latex]y={\mathrm{log}}_{b}\left(x\right),b>1[/latex], is

  • shifted vertically up [latex]d[/latex] units.
  • shifted horizontally to the left [latex]c[/latex] units.
  • stretched vertically by a factor of [latex]|a|[/latex] if [latex]|a| > 0[/latex].
  • compressed vertically by a factor of [latex]|a|[/latex] if [latex]0 < |a| < 1[/latex].
  • reflected about the [latex]x[/latex]–axis when [latex]a < 0[/latex].

For [latex]f\left(x\right)=\mathrm{log}\left(-x\right)[/latex], the graph of the parent function is reflected about the [latex]y[/latex]-axis.

What is the vertical asymptote of [latex]f\left(x\right)=-2{\mathrm{log}}_{3}\left(x+4\right)+5[/latex]?

What is the vertical asymptote of [latex]f\left(x\right)=3+\mathrm{ln}\left(x - 1\right)[/latex]?

In the example below, you’ll write a common logarithmic function for the graph shown. Remember that all the functions studied in this course possess the characteristic that every point contained on the graph of a function satisfies the equation of the function. As you have done before, begin with the form of a transformed logarithm function, [latex]f(x)=a\text{log}(x+c)+d[/latex], then fill in the parts you can discern from the graph.

  • Find the horizontal shift by locating the vertical asymptote.
  • Examine the shape of the graph to see if it has been reflected.
  • Once you have filled in what you know, substitute one or more points in integer coordinates if possible to solve for any remaining unknowns.
  • Remember that if there are more than one unknown, you’ll need more than one point and more than one equation to solve for all the unknowns.

Work through the example step-by-step with a pencil on paper, perhaps more than once or twice, to gain understanding.

Find a possible equation for the common logarithmic function graphed below.Graph of a logarithmic function with a vertical asymptote at x=-2, has been vertically reflected, and passes through the points (-1, 1) and (2, -1).

Give the equation of the natural logarithm graphed below.Graph of a logarithmic function with a vertical asymptote at x=-3, has been vertically stretched by 2, and passes through the points (-1, -1).