Logaritmisk

Logaritmisk is a term used to describe anything relating to logarithms, a class of functions that are the inverse of exponential growth. For a base b > 0 and b ≠ 1, the logarithm of a positive number x is the exponent y that satisfies b^y = x. This is written as log_b x, with common cases including the base 10 logarithm (log) and the natural logarithm (ln), which uses base e.

Key properties follow from the rules of exponents. 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, log_b x = log_k x / log_k b, allows conversion between bases. The function y = log_b x is defined for x > 0 and is monotone increasing if b > 1, or monotone decreasing if 0 < b < 1. It is the inverse of the exponential function y = b^x, and has domain x > 0 with range all real numbers. The derivative is d/dx log_b x = 1/(x ln b).

Applications of logarithmic concepts are widespread. They are used to transform multiplicative data into additive form, to construct logarithmic scales such as pH, decibels, and the Richter magnitude, and to measure information content (log base 2, in bits). In computer science, logarithms appear in algorithmic complexity analyses, and in calculus they appear in integrals and differentiations, for example, ∫(1/x) dx = ln|x| + C.

Historically, logarithms were introduced in the early 17th century by John Napier and further developed by Henry Briggs, providing a powerful tool for simplifying multiplication and division.