Exponential and Logarithmic Quick Review
- Convert between exponential and logarithmic forms to simplify calculations.
- Power Rule (Quirkle's property): If the base is the same, b^x = b^y implies x = y (for b>0, b\neq 1).
- Key equivalence: \log_b(x) = y \iff b^y = x.
- When stuck, convert to exponential form and solve by comparing exponents.
Quick Examples
- 2^4 = 16, so 16 = 2^4.
- \log_{10}\left(\frac{1}{100}\right) = -2 (since 10^{-2} = \frac{1}{100}).
- 3^4 = 81 \Rightarrow \log_3(81) = 4.
- Solve 4^y = 16 \Rightarrow 16 = 4^2 \Rightarrow y = 2.
- Solve \left(\frac{1}{2}\right)^y = 8
- Write base as powers: \frac{1}{2} = 2^{-1}, \; 8 = 2^3
- So 2^{-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.