Logarithmic Functions and Equations

Definition

$\log_b(x) = y \iff b^y = x$

Special cases:

• $\log(x) = \log_{10}(x)$
• $\ln(x) = \log_e(x)$

Properties of Logarithms

$\log_b(MN) = \log_b M + \log_b N$ $\log_b\left(\frac{M}{N}\right) = \log_b M - \log_b N$ $\log_b(M^p) = p \log_b M$ $\log_b b = 1 \quad \log_b 1 = 0$

Change of Base Formula

$\log_b x = \frac{\ln x}{\ln b} = \frac{\log x}{\log b}$

Graphs of Logarithmic Functions

$y = \log_b x$ is the inverse of $y = b^x$:

| Feature | $y = b^x$ | $y = \log_b x$ | |---------|----------|----------------| | Domain | $(-\infty, \infty)$ | $(0, \infty)$ | | Range | $(0, \infty)$ | $(-\infty, \infty)$ | | Asymptote | $y = 0$ | $x = 0$ | | Key point | $(0, 1)$ | $(1, 0)$ |

Solving Logarithmic Equations

Strategy 1: Convert to exponential form $\log_3(x+2) = 4 \implies x + 2 = 3^4 = 81 \implies x = 79$

Strategy 2: Combine logs, then convert $\ln x + \ln(x-2) = \ln 3$ $\ln[x(x-2)] = \ln 3$ $x^2 - 2x = 3 \implies x = 3 \quad (x = -1 \text{ extraneous})$

Solving Exponential Equations

$5^{2x-1} = 125 \implies 5^{2x-1} = 5^3 \implies 2x - 1 = 3 \implies x = 2$

$3^x = 20 \implies x = \frac{\ln 20}{\ln 3} \approx 2.727$

Semi-Log Plots

When data is plotted on a semi-log scale (log y vs. x), exponential data appears linear.

$\ln y = kt + \ln a \quad \text{(slope } = k, \text{ y-intercept } = \ln a\text{)}$

AP Precalculus Tip: Semi-log and log-log plots are emphasized on the exam. If data is linear on a semi-log plot, it's exponential.