Logaritmu

Logaritmu is the mathematical concept that expresses the inverse operation of exponentiation. For a base b > 0 with b ≠ 1, the logaritmu of a positive number y with base b is the exponent x such that b^x = y. It is written log_b(y), or simply log(y) when the base is understood from context. The base determines the properties and scale of the logarithm.

Common choices are base 10 (the common logarithm), base e (the natural logarithm, written ln), and base 2 (the binary logarithm, used in computer science). The change-of-base formula allows computing log_b(y) as log_k(y) / log_k(b) for any positive base k ≠ 1.

Key laws: log_b(xy) = log_b x + log_b y; log_b(x/y) = log_b x − log_b y; log_b(x^k) = k log_b x. The domain is x > 0; the graph is increasing if b > 1 and decreasing if 0 < b < 1. Derivative: d/dx log_b x = 1/(x ln b).

History and computation: Logarithms were developed in the early 17th century by John Napier and later extended by Henry Briggs to decimal bases. They enabled rapid multiplication and division via addition of logs. Before calculators, log tables and slide rules were essential tools.

Applications span science, engineering, and data analysis. Logarithms linearize multiplicative relationships, compress large ranges (as in the Richter scale or decibels), and appear in solving exponential growth/decay problems. In mathematics, they provide a bridge between algebra and calculus and underpin many numerical methods.