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Study Guides > College Algebra

Key Concepts

Key Equations

General Form for the Translation of the Parent Logarithmic Function  f(x)=logb(x)\text{ }f\left(x\right)={\mathrm{log}}_{b}\left(x\right)\\ f(x)=alogb(x+c)+d f\left(x\right)=a{\mathrm{log}}_{b}\left(x+c\right)+d\\

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

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