floorlogk

floorlogk is the floor of the logarithm of a positive number n in base k. Formally, floorlogk(n) = floor(log_k(n)), defined for n > 0 and k > 1. It returns the greatest integer m such that k^m <= n < k^(m+1). The function is nondecreasing in n and increases by 1 whenever n crosses a power of k.

For positive integers, floorlogk(n) is closely related to the base-k representation of n. The number of digits of n in base k is floorlogk(n) + 1. For example, floorlog2(1) = 0, floorlog2(2) = 1, floorlog2(3) = 1, floorlog10(1000) = 3, and floorlog3(8) = 1 because 3^1 <= 8 < 3^2.

Computing floorlogk(n) can be done via floating-point arithmetic as floor(log(n)/log(k)), though precautions are needed to avoid rounding errors. Integer methods exist as well, such as repeatedly dividing n by k to count how many times this is possible before reaching 0 or 1, or using fast integer logarithm algorithms.

Common applications include analysis of algorithms with logarithmic time or space complexity, estimation of the number of digits in a given base, and determining the size thresholds in data structures where growth occurs by a constant factor k. The concept generalizes to real n and to bases k in (0,1) with appropriate adjustments, though standard usage typically assumes k > 1 and n > 0.