Logaritmiga

Logaritmiga is the branch of mathematics that studies logarithmic functions and the transformations derived from them. It encompasses the definition, properties, and algebra of logarithms, as well as their use in modeling multiplicative relationships, data on logarithmic scales, and the linearization of exponential growth. In many contexts, logaritmiga also refers to the practical techniques for applying logarithms in science, engineering, and computation.

A logarithm with base b is defined for positive x as log_b(x), representing the exponent to which b must be raised to obtain x. The base b must be positive and not equal to 1. The logarithm is the inverse function of the exponential function b^x. Key domains and ranges are x > 0 and log_b(x) ∈ R. When b > 1, log_b(x) is increasing; when 0 < b < 1, it is decreasing. The derivative is 1/(x ln b), and the function has a vertical asymptote at x = 0.

Fundamental identities include 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 computations across different bases. Common bases are e (natural logarithm, ln) and 10 (common logarithm, log).

Applications of logaritmiga span data transformation to stabilize variance, analysis on logarithmic scales in science, and the study of algorithmic complexity (log n). It also underpins many engineering practices, such as decibel scales and pH measurement, where multiplicative relationships are converted to additive ones.

See also: logarithm, exponential function, logarithmic scale, change of base. References vary by language and discipline.

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