Logaxylogaxlogay

Logaxylogaxlogay is a neologistic term used in some mathematical puzzle and recreational mathematics contexts to denote a specific three-step nested logarithmic operation that uses three different bases.

Formally, for positive real numbers a, x, and y with a ≠ 1, and such that the nested arguments are defined, logaxylogaxlogay can be described by the expression L = log_y( log_x( log_a( x ) ) ). The inner term log_a(x) requires x > 0 and a as specified; to ensure log_x( log_a(x) ) is defined, log_a(x) must be positive, which imposes conditions on x relative to a; similarly, log_y( ...) requires y > 0 and y ≠ 1. The overall value is defined only on the domain where all three logarithms are defined.

Example: let a = 2, x = 4, y = 3. Then log_2(4) = 2; log_4(2) = 1/2; log_3(1/2) = ln(1/2)/ln(3) ≈ -0.631. This illustrates that the value can be negative even with positive inputs, due to the nested and cross-base structure.

Origin and usage: The term is not standard in formal mathematics and appears mainly in informal writings and puzzles as a mnemonic or playful label for the triple-nested construction. It can be used to explore base-change behavior and the sensitivity of nested logarithms to input values.

See also: logarithm, iterated logarithm, base change formula; related ideas include nested functions and composition in elementary functions.