Properties of Logarithms

The three log rules you must memorize

Product rule: $\log(ab) = \log a + \log b$ — multiplication becomes addition.

Quotient rule: $\log(a/b) = \log a - \log b$ — division becomes subtraction.

Power rule: $\log(a^n) = n \log a$ — exponents become coefficients.

Example: Expand $\log_3(x^2 y / z^3)$

Step 1 — Quotient rule: $\log_3(x^2 y) - \log_3(z^3)$

Step 2 — Product rule: $\log_3(x^2) + \log_3(y) - \log_3(z^3)$

Step 3 — Power rule: $2\log_3 x + \log_3 y - 3\log_3 z$

Going backward (condensing)

$3\log x - \log(x+1) = \log\left(\frac{x^3}{x+1}\right)$

These rules are essential for solving logarithmic equations and simplifying expressions. They convert multiplication into addition, division into subtraction, and exponentiation into multiplication. These rules make logarithms powerful tools for simplifying calculations.