Logxyk

Logxyk is a term that appears in informal or niche mathematical discussions to denote the k-fold logarithm of a value y with base x. It is not a standard, universally adopted notation in mainstream mathematics, but it is sometimes used to discuss iterated logarithmic operations in a concise way.

Definition and notation. For a base x with x > 0 and x ≠ 1, and a positive y, define L1 as log base x of y, denoted log_x(y). Then for n ≥ 1, set L_{n+1} = log_x(L_n). The expression logxyk refers to L_k, provided that all intermediate values L_1, L_2, ..., L_k are defined (each must be positive to serve as an input to the next logarithm). If at any stage L_n ≤ 0, the next logarithm is undefined, so logxyk is only defined up to a maximal k determined by x and y.

Behavior and considerations. The iterated logarithm tends to decrease rapidly as k increases when x > 1 and y > 1, but the operation is only defined while all intermediate arguments remain positive. For bases 0 < x < 1, the monotonic behavior changes, and the iteration can become undefined at an even earlier stage. Because logxyk depends on both x and y, there is no general closed-form expression for arbitrary k; each case is typically treated by explicit computation or by qualitative analysis of the recurrence L_{n+1} = log_x(L_n).

Relation and context. Logxyk is related to the broader ideas of iterated logarithms and nested logarithmic expressions, and is sometimes discussed alongside concepts such as the iterated logarithm function log* n and tetration. It serves mainly as a compact descriptive gadget in problem-solving and expository writing rather than a standardized mathematical operator.