Logarithm

A logarithm is the inverse function of exponentiation with a fixed base. For a base b > 0 and b ≠ 1, the logarithm of x > 0 is the exponent y such that b^y = x. This is written log_b(x). The natural logarithm uses base e and is denoted ln x; the common logarithm uses base 10 and is often written as log x.

Key properties include log_b(1) = 0 and log_b(b) = 1. The logarithm is increasing if b > 1 and decreasing if 0 < b < 1. Core identities are log_b(xy) = log_b(x) + log_b(y); log_b(x^k) = k log_b(x); and log_b(x/y) = log_b(x) − log_b(y). The change of base formula allows computation in any base: log_b(x) = log_k(x) / log_k(b) for any positive base k ≠ 1.

Domain considerations: The logarithm is defined for x > 0 in the real numbers. Logs of nonpositive numbers are not real-valued. Bases must satisfy b > 0 and b ≠ 1; otherwise the function is not a logarithm.

Applications and history: Logarithms simplify multiplication and division through addition of logarithms, and they solve exponential equations. They are used in modeling growth and decay, data transformation, and various scientific scales such as pH, decibels, and the Richter scale. Historically, John Napier introduced logarithms in the early 17th century; Henry Briggs developed the base-10 (common) logarithms, and the natural logarithm with base e is central to calculus, often denoted ln.