logxn

Logxn is a compact notation that is sometimes used to denote the logarithm of a number x with base n. In standard mathematical writing this quantity is usually written as log base n of x, log_n(x). The form logxn can appear in calculators, programming libraries, or informal text, but its meaning can vary by source, since some authors use log_x(n) to mean the logarithm of n with base x.

The function log_n(x) is defined for x > 0 with a base n satisfying n > 0 and n ≠ 1. If n > 1, the function is increasing; if 0 < n < 1, it is decreasing. It satisfies log_n(1) = 0 and log_n(n) = 1. Standard rules include log_n(xy) = log_n(x) + log_n(y), log_n(x^k) = k log_n(x), and log_n(x/y) = log_n(x) − log_n(y). A key relation is the change-of-base formula: log_n(x) = log_k(x) / log_k(n) for any positive base k ≠ 1.

Computing log_n(x) in practice often uses a more basic logarithm such as the natural log or base-10 log: log_n(x) = ln(x) / ln(n) = log_10(x) / log_10(n). This makes it easy to evaluate with calculators or software that provide logs in a single base.

Examples illustrate the concepts: log_2(8) = 3, since 2^3 = 8; log_{1/2}(4) = −2, since (1/2)^(−2) = 4. Logarithms are used in scaling, exponential growth analysis, and information theory, where different bases reflect different measurement units. See also logarithm, change of base, and natural logarithm.