Key Equations

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 $f\left(x\right)={\mathrm{log}}_{b}\left(x\right)$ has an x-intercept at $\left(1,0\right)$, domain $\left(0,\infty \right)$, range $\left(-\infty ,\infty \right)$, vertical asymptote x = 0, and
  • if b > 1, the function is increasing.
  • if 0 < b < 1, the function is decreasing.
• The equation $f\left(x\right)={\mathrm{log}}_{b}\left(x+c\right)$ shifts the parent function $y={\mathrm{log}}_{b}\left(x\right)$ horizontally
  • left c units if c > 0.
  • right c units if c < 0.
• The equation $f\left(x\right)={\mathrm{log}}_{b}\left(x\right)+d$ shifts the parent function $y={\mathrm{log}}_{b}\left(x\right)$ vertically
  • up d units if d > 0.
  • down d units if d < 0.
• For any constant a > 0, the equation $f\left(x\right)=a{\mathrm{log}}_{b}\left(x\right)$
  • stretches the parent function $y={\mathrm{log}}_{b}\left(x\right)$ vertically by a factor of a if |a| > 1.
  • compresses the parent function $y={\mathrm{log}}_{b}\left(x\right)$ vertically by a factor of a if |a| < 1.
• When the parent function $y={\mathrm{log}}_{b}\left(x\right)$ 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 $f\left(x\right)=-{\mathrm{log}}_{b}\left(x\right)$ represents a reflection of the parent function about the x-axis.
  • The equation $f\left(x\right)={\mathrm{log}}_{b}\left(-x\right)$ 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 $f\left(x\right)=a{\mathrm{log}}_{b}\left(x+c\right)+d$.
• Given an equation with the general form $f\left(x\right)=a{\mathrm{log}}_{b}\left(x+c\right)+d$, we can identify the vertical asymptote x = –c for the transformation.
• Using the general equation $f\left(x\right)=a{\mathrm{log}}_{b}\left(x+c\right)+d$, we can write the equation of a logarithmic function given its graph.