logaritmave

Logaritmave, commonly known as logarithms, are inverse functions of exponentiation. For a base b > 0 with b ≠ 1, the logarithm of a positive number x is written log_b(x) and is defined as the unique exponent y such that b^y = x. In other words, log_b(x) tells you the power to which you must raise b to obtain 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 evaluation with any positive base k ≠ 1. Bases commonly used are the natural base e, giving the natural logarithm ln(x), and base 10, giving the common logarithm log_10(x). For bases b > 1, log_b is increasing; for 0 < b < 1, it is decreasing. The domain of log_b is x > 0 and its range is all real numbers.

In the complex setting, logarithms become multi-valued: log(z) = ln|z| + i Arg(z), where Arg(z) is multi-valued due to the periodicity of the complex exponential.

Historically, logarithms were introduced in the early 17th century by John Napier to simplify multiplication and division, later developed with base-10 by Henry Briggs. They remain fundamental in science and engineering for solving exponential equations, transforming data across scales, and in fields such as information theory and computer science, where logarithms describe growth rates, data compression, and algorithm complexity.