Precalculus / Unit 3: Exponential and Logarithmic Functions
3.3 Properties of Logarithms and Change of Base
Notes
For positive $M$, $N$ and valid base $b$:
- Product: $\log_b (MN) = \log_b M + \log_b N$.
- Quotient: $\displaystyle \log_b \frac{M}{N} = \log_b M - \log_b N$.
- Power: $\log_b M^p = p \log_b M$.
- Change of base: $\displaystyle \log_b x = \frac{\ln x}{\ln b} = \frac{\log x}{\log b}$.
- Example: $\displaystyle \log_8 32 = \frac{\log_2 32}{\log_2 8} = \frac{5}{3}$.
Practice
Simplify $\log_6 4 + \log_6 9$.
The product rule gives $\log_6 36 = 2$.
Expand $\displaystyle \log_3 \frac{x^2}{y}$.
Quotient rule turns division into subtraction; power rule brings down the 2.
Which equals $\log_8 32$?
Change of base: $\displaystyle \frac{\log_2 32}{\log_2 8} = \frac{5}{3}$.
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