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Finding the Inverse of Logarithmic and Exponential Equations

  • Substitute y for f(x)
  • switch x and y
  • solve for y
  • substitute f^{-1}(x) for y
    • f^{-1}(x) is the inverse

Examples

  1. f(x)=e^x
    • y=e^x
    • x=e^y
    • \ln{(x)}=\ln{(e^y)}
      • Note: \ln{(e^y)}=y
    • \ln{(x)}=y
    • f^{-1}(x)=\ln(x)
  2. f(x)=ln(x+y)
    • y=ln(x+5)
    • x=ln(y+5)
    • e^{ln(x)}-x
    • e^x=e^{ln(y+5)}
    • e^x=y+5
    • -5          -5
    • e^x-5=y
    • f^{-1}(x)=e^x-5
  3. f(x)=ln(x+5)
    • y=ln(x+5)
    • x=ln(y+5)
    • e^{\ln(x)}-x
    • e^x=e^{ln(y+4)}
    • e^x=y+5
    • -5           -5
    • e^x-5=y
    • f^{-1}(x)=e^x-5
  4. f(x)=10^{x-1}
    • y=10^{x-1}
    • x=10^{y-1}
    • \log(x)=\log(10^{y-1})\cdot \log(10^x)=x
    • log(x)=y-1
    • +1          +1
    • log(x)+1=y
    • f^{-1}(x)=\log(x)+1