Indices and Logarithms - I — O/L Mathematics

Indices (also called exponents or powers) provide a shorthand way of writing repeated multiplication. For any real number a and a positive integer n, aⁿ means a multiplied by itself n times. The number a is called the base, and n is the index (or exponent). For example, 2³ = 2 × 2 × 2 = 8.

The concept of indices extends beyond positive integers. We define: a⁰ = 1 for any a ≠ 0. This makes sense because dividing a^m by a^m gives a^m-m = a⁰ = 1. a^-n = 1/(aⁿ) for a ≠ 0. This extends the pattern: a³ ÷ a⁵ = a^-2 = 1/a². a^1/n = the nth root of a, written as √[n]a. For example, 8^1/3 = ∛8 = 2. a^m/n = (a^1/n)^m = (a^m)^1/n, meaning the nth root of a^m.

Logarithms are the inverse operation of indices. If a^b = c, then we say the logarithm of c to base a is b, written as log_a c = b. Here a > 0, a ≠ 1, and c > 0. For example, since 2³ = 8, we have log₂ 8 = 3. The logarithm answers the question: "To what power must the base be raised to get the given number?"