Logarithms and Their Properties

Definition of Logarithm

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

"Log base $b$ of $x$ equals $y$" means "$b$ to the $y$ power equals $x$."

Examples: $\log_2(8) = 3 \quad \text{because} \quad 2^3 = 8$ $\log_5(25) = 2 \quad \text{because} \quad 5^2 = 25$ $\log_{10}(1000) = 3 \quad \text{because} \quad 10^3 = 1000$

Common and Natural Logarithms

• $\log x = \log_{10} x$ (common log)
• $\ln x = \log_e x$ (natural log, $e \approx 2.718$)

Properties of Logarithms

Product Rule

$\log_b(MN) = \log_b M + \log_b N$

Quotient Rule

$\log_b\left(\frac{M}{N}\right) = \log_b M - \log_b N$

Power Rule

$\log_b(M^p) = p \cdot \log_b M$

Change of Base

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

Special Values

$\log_b 1 = 0 \quad \log_b b = 1 \quad \log_b b^x = x \quad b^{\log_b x} = x$

Solving Logarithmic Equations

Example 1: $\log_2(x-1) = 4$ $x - 1 = 2^4 = 16 \implies x = 17$

Example 2: $\log x + \log(x+3) = 1$ $\log[x(x+3)] = 1 \implies x(x+3) = 10$ $x^2 + 3x - 10 = 0 \implies (x+5)(x-2) = 0$ $x = 2 \quad (x = -5 \text{ is extraneous — can't log a negative})$

Solving Exponential Equations with Logs

$3^{2x} = 15$ $2x \cdot \ln 3 = \ln 15$ $x = \frac{\ln 15}{2 \ln 3} \approx 1.232$

Graphs

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

• Domain: $x > 0$
• Range: All real numbers
• Vertical asymptote: $x = 0$
• Passes through $(1, 0)$ and $(b, 1)$

Key relationship: Logarithms and exponentials are INVERSES. If you're stuck, convert between forms!