xlogk

xlogk is a two-argument mathematical function defined by f(x, k) = x log_k x, where x > 0 and k > 0 with k ≠ 1. The logarithm is taken in base k, and log_k x can be expressed as ln x / ln k, so equivalently f(x, k) = x ln x / ln k. The form is a variant of the familiar x log x and differs only by a constant factor depending on the base k.

Basic properties: Since log_k x = ln x / ln k, the function inherits many characteristics of x ln x up to a constant. The derivative is f'(x, k) = (ln x + 1) / ln k. Thus the sign of the slope depends on k: for k > 1 (ln k > 0), f is increasing for x > e^{-1} and, overall, grows with x; for 0 < k < 1 (ln k < 0), the sign is reversed. As x approaches 0 from the right, f(x, k) tends to 0; as x grows large, f(x, k) grows roughly like (1/ln k) x ln x.

Asymptotics and related notions: Because log_k x = log x / log k, x log_k x = (x log x) / log k, the asymptotic growth is Θ(x log x) for any fixed k > 0, k ≠ 1. This makes xlogk a commonly used notation in analyses where a constant-base logarithm is involved, such as in algorithmic time or space bounds where the base change does not affect the overall growth rate.

Example: with x = 8 and k = 2, f(8, 2) = 8 log_2 8 = 8 × 3 = 24.