Summary: Graphs of Logarithmic Functions

Key Equations

General Form for the Transformation of the Parent Logarithmic Function [latex]\text{ }f\left(x\right)={\mathrm{log}}_{b}\left(x\right)[/latex] [latex]f\left(x\right)=a{\mathrm{log}}_{b}\left(x+c\right)+d[/latex]

Key Concepts

  • 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.
  • The equation [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x+c\right)[/latex] shifts the parent function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] horizontally
    • left c units if c > 0.
    • right c units if c < 0.
  • The equation [latex]f\left(x\right)={\mathrm{log}}_{b}\left(x\right)+d[/latex] shifts the parent function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] vertically
    • up d units if d > 0.
    • down d units if d < 0.
  • For any constant a > 0, the equation [latex]f\left(x\right)=a{\mathrm{log}}_{b}\left(x\right)[/latex]
    • stretches the parent function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] vertically by a factor of a if |a| > 1.
    • compresses the parent function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] vertically by a factor of a if |a| < 1.
  • When the parent function [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex] is multiplied by –1, the result is a reflection about the x-axis. When the input is multiplied by –1, the result is a reflection about the y-axis.
    • The equation [latex]f\left(x\right)=-{\mathrm{log}}_{b}\left(x\right)[/latex] represents a reflection of the parent function about the x-axis.
    • The equation [latex]f\left(x\right)={\mathrm{log}}_{b}\left(-x\right)[/latex] represents a reflection of the parent function about the y-axis.
    • A graphing calculator may be used to approximate solutions to some logarithmic equations.
  • All transformations of the logarithmic function can be summarized by the general equation [latex]f\left(x\right)=a{\mathrm{log}}_{b}\left(x+c\right)+d[/latex].
  • Given an equation with the general form [latex]f\left(x\right)=a{\mathrm{log}}_{b}\left(x+c\right)+d[/latex], we can identify the vertical asymptote x = –c for the transformation.
  • Using the general equation [latex]f\left(x\right)=a{\mathrm{log}}_{b}\left(x+c\right)+d[/latex], we can write the equation of a logarithmic function given its graph.