3.2 Logarithms as Inverses

Notes

A logarithm answers "what exponent?": $\log_b x = y$ means $b^y = x$ (with $b > 0$, $b \neq 1$, $x > 0$).

• $\log_2 8 = 3$ because $2^3 = 8$. $\displaystyle \log_5 \frac{1}{25} = -2$ because $\displaystyle 5^{-2} = \frac{1}{25}$.
• Common log: $\log x = \log_{10} x$. Natural log: $\ln x = \log_e x$ where $e \approx 2.718$.
• $\log_b 1 = 0$ and $\log_b b = 1$.
• $y = \log_b x$ and $y = b^x$ are inverse functions, so $\ln(e^4) = 4$ and $10^{\log 7} = 7$.
• The domain of $\log(x - 3)$ is $x > 3$.