logaritm

Logaritm, or logarithm, is a mathematical function that serves as the inverse of exponentiation. For a base b with b > 0 and b ≠ 1, the logarithm of a positive number x with base b is the exponent y such that b^y = x. It is denoted as log_b(x).

Common bases are base 10, base e, and base 2. The base 10 logarithm is often called the common logarithm and is written log(x). The base e logarithm is the natural logarithm and is written ln(x). The base 2 logarithm is the binary logarithm, written log2(x).

The logarithm is defined for x > 0. If the base b > 1, log_b is an increasing function; if 0 < b < 1, it is decreasing. Key values include log_b(1) = 0 and log_b(b) = 1. Important rules include log_b(xy) = log_b(x) + log_b(y) and log_b(x^k) = k log_b(x). The change of base formula allows computation with any base: log_b(x) = ln(x)/ln(b) = log_k(x)/log_k(b).

Graphically, a logarithm curve passes through (1, 0) and increases (or decreases) without bound as x grows, with a vertical asymptote at x → 0+. It is defined only for positive x, reflecting its role as the exponent that produces x from the base.

Applications of logarithms span science and engineering, where they convert multiplicative relationships to additive scales, aid in data compression, and support algorithms handling large ranges of values. The concept originated in the early 17th century with John Napier and was later popularized for base 10 by Henry Briggs.