logarithmiskt

Logarithmiskt describes mathematical concepts and functions that involve logarithms, the inverse of exponentiation. For a positive number x and a base b with b > 0 and b ≠ 1, the logarithm of x to base b is the exponent y that satisfies b^y = x. This is written as log_b(x). The natural logarithm uses base e and is written ln(x); the common logarithm uses base 10 and is written log(x).

Key properties include log_b(1) = 0 and log_b(b) = 1. The product rule log_b(xy) = log_b(x) + log_b(y) and the power rule log_b(x^k) = k log_b(x) hold for positive x. The change of base formula log_b(x) = log_k(x) / log_k(b) allows converting between bases. Logarithms are the inverse functions of exponential functions: if y = log_b(x), then b^y = x. The domain requires x > 0, and the base must satisfy b > 0 and b ≠ 1. For bases b > 1, log_b(x) is increasing; for 0 < b < 1, it is decreasing. Graphs of logarithmic functions have a vertical asymptote at x = 0 and pass through (1, 0).

Historically, logarithms were developed to simplify multiplication and division by converting them into addition and subtraction. They underpin logarithmic scales, data transformations, and various scientific and engineering applications. In practice, natural logarithms are common in calculus and analysis, while base-10 logarithms appear in many measurement contexts. Logarithms remain a fundamental tool in mathematics, science, and data analysis.