Precalculus / Unit 3: Exponential and Logarithmic Functions
3.2 Logarithms as Inverses
Notes
A logarithm answers "what exponent?": $\log_b x = y$ means $b^y = x$.
- $\log_b 1 = 0$ and $\log_b b = 1$.
- $\log x$ means $\log_{10} x$. $\ln x$ means $\log_e x$.
- The domain of $\log_b x$ is $x > 0$.
- Examples: $\log_2 8 = 3$ because $2^3 = 8$; $\displaystyle \log_5 \frac{1}{25} = -2$; $\displaystyle \log_9 3 = \frac{1}{2}$.
Practice
Evaluate $\log_4 64$.
$4^3 = 64$.
Evaluate $\displaystyle \log_2 \frac{1}{8}$.
$\displaystyle 2^{-3} = \frac{1}{8}$.
Evaluate $\log_9 27$.
$9^x = 27$ means $3^{2x} = 3^3$, so $2x = 3$ and $\displaystyle x = \frac{3}{2}$.
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