Logaritmseks

Logaritmseks is a neologism encountered in informal mathematical writing and some fiction to describe a class of transformations that combines logarithmic scaling with piecewise linear behavior in log space. It is not a standard term in formal mathematics.

In the simplest informal definition, a logaritmseks transformation T acts on positive real numbers x > 0 by first computing u = log_b x, for a chosen base b > 1, and then applying a piecewise linear function f to u: T(x) = f(log_b x). The function f is continuous and built from a finite number of linear segments with slopes that may differ across the segments, producing a map whose output on a log scale is piecewise linear.

If all slopes of f are positive, T is monotone; if f is strictly increasing, T is invertible on its domain. The transformation is inherently tied to the chosen base b; scaling x by a constant translates u by a constant, which translates into a predictable shift in T when f is linear on the corresponding segment. Because the construction depends on log space, T inherits the scale-aware behavior of logarithms and can model data that change at different rates across ranges of x.

Take b=10 and f(u) = 2u for u ≤ 1, and f(u) = u + 1 for u > 1, with continuity at u=1. Then T(x) behaves as 2 log10 x for small x and as log10 x plus 1 for larger x, illustrating a two-slope log-scale model.

Logaritmseks is mainly of pedagogical interest or used in speculative writing to illustrate how log scales can interact with segmentation. It is not part of standard mathematical theory, and different authors may define it differently.

See also: logarithm, log-scale, piecewise linear function, log-log plot.