logarithmische

Logarithmische, in English logarithmic, refers to concepts and phenomena associated with logarithms in mathematics. A logarithm of a positive number x with base b greater than zero and not equal to one is the exponent y that satisfies b^y = x. It is written as log_b(x). Common bases are e (the natural logarithm, denoted ln x) and 10 (the common logarithm, denoted log x), while base 2 is often used in computer science.

Key properties include log_b(xy) = log_b x + log_b y, log_b(x/y) = log_b x − log_b y, and log_b(x^k) = k log_b x. The change of base formula, log_b x = log_k x / log_k b, allows conversion between bases. The logarithm is the inverse function of exponentiation: b^{log_b x} = x and log_b(b^x) = x. The domain for logarithms requires x > 0, and the base b must satisfy b > 0, b ≠ 1. Derivatives and integrals are used in analysis: d/dx log_b x = 1/(x ln b) and ∫ log_b x dx = x log_b x − x/ln b + C.

Logarithms underpin many scales and applications. They enable data to be compressed or linearized on a log scale, as in the Richter scale for earthquakes, pH, decibels, and certain biological and economic measurements. They also determine algorithmic complexity, where many processes have time that grows logarithmically with input size. The concept emerged from calculations attributed to Napier and was refined with Briggs and the constant e, yielding the natural logarithm central to analysis.