Logaritmik

Logaritmik is an adjective describing anything related to logarithms, a family of functions that are the inverse of exponentiation. The logarithm of a positive number x with respect to a base b > 0 and b ≠ 1 is the exponent y such that b^y = x. The natural logarithm uses base e and is denoted ln x; the common logarithm uses base 10 and is often written log x.

Key identities include log_b(xy) = log_b x + log_b y and log_b(x^k) = k log_b x. The change of base formula is log_b x = log_k x / log_k b for any positive base k ≠ 1. The domain for logarithms is x > 0.

Logaritmik scales are used to represent data spanning several orders of magnitude, compressing large ranges and turning multiplicative relationships into additive ones. They appear in science and engineering with examples such as the decibel scale for sound, the pH scale for acidity, and the Richter scale for earthquakes. In calculus, the natural log satisfies d/dx ln x = 1/x and ∫(1/x) dx = ln|x| + C.

In computer science, logarithmic time complexity, written O(log n), describes algorithms whose running time grows slowly with input size. Logarithms also feature in statistics, such as log-transformations used to stabilize variance or normalize skewed distributions. Historically, logarithms were introduced in the early 17th century by John Napier and popularized by Henry Briggs, enabling multiplication to be performed as addition.