Indices and Logarithms - I

Indices (also called exponents or powers) provide a shorthand way of writing repeated multiplication. For any real number a and a positive integer n, a^n 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/(a^n) 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 loga 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?"