Algebra 2 / Unit 5: Exponential and Logarithmic Functions
5.3 Logarithms
Notes
A logarithm is an exponent. $\log_b x = y$ means $b^y = x$.
- Read $\log_2 8$ as "2 to what power gives 8?" The answer is 3.
- Common log: $\log x$ means $\log_{10} x$. Natural log: $\ln x$ means $\log_e x$.
- Key facts: $\log_b 1 = 0$, $\log_b b = 1$, $\log_b b^x = x$, $b^{\log_b x} = x$.
- You can only take the log of a positive number.
- Example: $\displaystyle \log_3 \frac{1}{9} = -2$ because $\displaystyle 3^{-2} = \frac{1}{9}$.
Practice
Evaluate $\log_2 32$.
$2^5 = 32$.
Write $3^4 = 81$ in logarithmic form.
The base stays the base; the log equals the exponent.
Evaluate $\displaystyle \log_5 \frac{1}{25}$.
$\displaystyle 5^{-2} = \frac{1}{25}$.
Ready for more? Take the Unit 5 test.