logaritme

Logaritme (logarithm) is a mathematical function that serves as the inverse of exponentiation. For a base b > 0 with b ≠ 1, the logarithm of a positive number x is the exponent y such that b^y = x. This is written log_b(x) = y. Consequently, log_b(b^x) = x and b^{log_b(x)} = x.

Common bases are base 10 (the common logarithm, written log), base e (the natural logarithm, written ln), and base 2 (the binary logarithm). The base is often omitted when the context is clear. Key rules include log_b(xy) = log_b x + log_b y; log_b(x^k) = k log_b x; log_b(x/y) = log_b x − log_b y; and the change of base formula log_b x = log_k x / log_k b for any base k > 0, k ≠ 1.

Domain and range: the domain is x > 0, and the range is all real numbers. The graph is increasing for b > 1 and decreasing for 0 < b < 1, with a vertical asymptote at x = 0 and the point (1, 0).

Applications: logarithms are used to solve exponential equations and model growth and decay. They transform multiplicative data into additive form, and appear in scales such as the common logarithm, natural logarithm, decibels, and pH. In computer science, logarithms measure algorithmic complexity (log n). In finance and science, they assist in compound interest calculations and data analysis across wide ranges.

History: logarithms were introduced by John Napier in 1614 to simplify multiplication, with Henry Briggs later promoting base-10 logarithms. The natural logarithm (base e) arises naturally in calculus and analysis, where e ≈ 2.71828 is a fundamental constant.