Exponential and Logarithmic Quick Review

Exponential Form and Logarithms

  • Convert between exponential and logarithmic forms to simplify calculations.
  • Power Rule (Quirkle's property): If the base is the same, bx=byb^x = b^y implies x=yx = y (for b>0,b≠1b>0, b\neq 1).
  • Key equivalence: log⁡b(x)=y  ⟺  by=x\log_b(x) = y \iff b^y = x.
  • When stuck, convert to exponential form and solve by comparing exponents.

Quick Examples

  • 24=162^4 = 16, so 16=2416 = 2^4.
  • log⁡10(1100)=−2\log_{10}\left(\frac{1}{100}\right) = -2 (since 10−2=110010^{-2} = \frac{1}{100}).
  • 34=81⇒log⁡3(81)=43^4 = 81 \Rightarrow \log_3(81) = 4.
  • Solve 4y=16⇒16=42⇒y=24^y = 16 \Rightarrow 16 = 4^2 \Rightarrow y = 2.
  • Solve (12)y=8\left(\frac{1}{2}\right)^y = 8
    • Write base as powers: 12=2−1,  8=23\frac{1}{2} = 2^{-1}, \; 8 = 2^3
    • So 2−y=23⇒−y=3⇒y=−32^{-y} = 2^3 \Rightarrow -y = 3 \Rightarrow y = -3.

Practice habit

  • Write the full steps to convert and solve; this builds a habit for when you cannot see the answer immediately.
  • If you can\'t see it, rely on the method (trust your instrument) and solve step by step rather than guessing.