Integrated Math 3 / Unit 3: Exponential and Logarithmic Functions
3.3 Properties of Logarithms
Notes
For $b > 0$, $b \neq 1$, $M, N > 0$:
- 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: $\displaystyle \log_b x = \frac{\log x}{\log b} = \frac{\ln x}{\ln b}$.
- Common mistake: $\log (M + N)$ is not $\log M + \log N$.
Practice
Write $\log_2 5 + \log_2 6$ as a single logarithm.
The product property: $\log_2 5 + \log_2 6 = \log_2 (5 \cdot 6)$.
Expand $\displaystyle \log \frac{x^2 y}{z^3}$.
Use the product, quotient and power properties.
Use the change of base formula to estimate $\log_2 20$.
$\displaystyle \frac{\log 20}{\log 2} \approx \frac{1.301}{0.301} \approx 4.32$. Check: $2^4 = 16$ and $2^5 = 32$.
Ready for more? Take the Unit 3 test.