Algebra 2 / Unit 5: Exponential and Logarithmic Functions
5.4 Properties of Logarithms
Notes
- Product: $\log_b (MN) = \log_b M + \log_b N$.
- Quotient: $\displaystyle \log_b \frac{M}{N} = \log_b M - \log_b N$.
- Power: $\log_b M^k = k \log_b M$.
- Change of base: $\displaystyle \log_b x = \frac{\ln x}{\ln b}$ (or use $\log$ on top and bottom).
- Example: $\log_2 12 - \log_2 3 = \log_2 4 = 2$.
- Watch out: $\log(M + N)$ does NOT equal $\log M + \log N$.
Practice
Expand $\displaystyle \log \frac{x^3 y}{z}$.
Product, quotient, then power rule on $x^3$ only.
Evaluate $2\log_3 6 - \log_3 4$.
$\log_3 36 - \log_3 4 = \log_3 9 = 2$.
Use change of base to approximate $\log_4 20$.
$\displaystyle \frac{\ln 20}{\ln 4} \approx \frac{2.996}{1.386} \approx 2.161$.
Ready for more? Take the Unit 5 test.