Key Equations
| General Form for the Translation 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 translations 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.