Floorlogkn

Floorlogkn, or floor log base k of n, denotes the greatest integer m such that k^m ≤ n. It is commonly written as floor_log_k(n). The definition requires k > 0 and k ≠ 1 and n > 0; in practice, most discussions assume k > 1. Under these conditions floor_log_k(n) is a nondecreasing function of n when k > 1, and it measures the size of n in base k.

Interpretations and relationships: floor_log_k(n) equals the integer part of the logarithm of n to base k. It is related to the number of digits of n in base k: digits(n, k) = floor_log_k(n) + 1 for n ≥ 1. It also satisfies floor_log_k(n) = floor( log_e(n) / log_e(k) ).

Computing: In floating-point arithmetic, m = floor( ln(n) / ln(k) ). In integer arithmetic without floating point, one can repeatedly divide n by k to count how many times k fits into n: initialize m = 0, t = n; while t ≥ k, t := floor(t / k), m := m + 1. This yields the same result as the mathematical definition.

Examples: floor_log_2(16) = 4; floor_log_10(500) = 2; floor_log_2(1) = 0. Edge cases: for n ≤ 0 the function is undefined in the usual real-valued setting; for k ≤ 0 or k = 1 the logarithm is undefined. When 0 < k < 1 the function is defined but decreases with n, and the floor introduces additional steps.