floorlognlogk

Floorlognlogk is a mathematical function defined as the floor of the logarithm with base n of the logarithm with base k of its argument. Formally, for real numbers n > 1, k > 1 and x > 1, floorlognlogk(x) = floor( log_n( log_k x ) ). Here floor denotes the greatest integer less than or equal to the argument.

Domain and range

- Domain: x > 1. This ensures log_k x is positive, so log_n(log_k x) is defined for n > 1.

- Range: The set of integers, since the floor operation yields integers.

Basic properties

- Monotonicity: floorlognlogk(x) is strictly increasing in x for x > 1, because both log_k x and log_n(·) are increasing functions when n, k > 1.

- Change points: The value of floorlognlogk(x) increases by 1 when log_k x reaches n^m for integers m, equivalently when x reaches k^{n^m}.

- Limiting behavior: As x approaches 1 from the right, floorlognlogk(x) tends to negative infinity. As x grows without bound, floorlognlogk(x) tends to positive infinity, but very slowly.

Relations and variations

- Analytical form: log_n( log_k x ) = [ln( ln x / ln k )] / ln n = (ln ln x − ln ln k) / ln n.

- Approximation: floorlognlogk(x) is approximately (ln ln x) / ln n plus a constant offset, then rounded down.

- Variants: If a related function uses different bases or omits the floor, it becomes log_n( log_k x ) or log log x scaled by log bases accordingly.

Applications

- Appears in analyses of algorithms and data structures where bounds involve nested logarithms, often in the context of extremely slowly growing quantities.

- Useful as a compact descriptor of double-logarithmic behavior, especially in theoretical computer science and number theory discussions.