Logbfx

Logbfx is a notational convention used in mathematics and computing to denote the logarithm of a function f(x) with a specified base b. In informal or compact notation, logbfx is often used as shorthand for log_b f(x). The expression log_b f(x) is defined for all x where f(x) > 0, with base b > 0 and b ≠ 1. It is related to the natural logarithm via log_b f(x) = ln(f(x)) / ln(b). The base b determines the scale of the logarithm: for b > 1 the function log_b is increasing in f(x); for 0 < b < 1 it is decreasing.

Domain and range: the domain consists of all x such that f(x) > 0; the range depends on f and b and is typically unbounded.

Differentiation and integration: if f is differentiable, d/dx log_b f(x) = f'(x) / (f(x) ln b). The integral ∫ log_b f(x) dx has no simple general form and depends on f.

In computing, log_b f(x) is often evaluated using a change of base: log_b f(x) = log f(x) / log b, where log denotes any convenient logarithm (natural, base 10, etc.). Some programming languages provide a direct logb function or require language-specific syntax.

Applications: log_b f(x) appears in growth and decay models, data transformation, information theory, and solving equations where multiplicative relationships are present.

History: The broader idea of logarithms with arbitrary bases extends the work of historical log tables; the base-change formula is a standard result in algebra.

See also: natural logarithm, common logarithm, base change, logarithmic transformation.