Laws of Exponents and Logarithms

From repeated multiplication to log scales

Exponents describe repeated multiplication, and logarithms are their inverses—the tool for "undoing" an exponent. Together they underpin exponential growth models and the logarithmic scales scientists use to tame enormous ranges of data, such as sound intensity, acidity, and earthquake energy.

Laws of exponents

For any nonzero base $a$ and rational exponents,

$$a^m \times a^n = a^{m+n}, \qquad \frac{a^m}{a^n} = a^{m-n}, \qquad (a^m)^n = a^{mn},$$ $$a^0 = 1, \qquad a^{-n} = \frac{1}{a^n}, \qquad a^{1/n} = \sqrt[n]{a}.$$

Worked example. Simplify $\dfrac{3^5 \times 3^{-2}}{3^{2}}$. Combining exponents: $3^{5 + (-2) - 2} = 3^{1} = 3.$ A fractional exponent gives roots: $16^{3/4} = \left(16^{1/4}\right)^3 = 2^3 = 8.$

Definition of a logarithm

The logar