Pseudologarithm

Pseudologarithm is not a single, formal mathematical object. In many fields it is used informally to describe a function that mimics a logarithm in key ways but is defined in a way that extends or modifies the logarithm. Such a function is typically monotone, unbounded, and behaves like a log on a chosen domain or asymptotically for large inputs, while avoiding domain restrictions or improving numerical properties.

Common realizations include shifted or generalized logarithms, such as f(x) = log(1+x) for x > -1, which extends the log to zero, and inverse hyperbolic sine, asinh(x), which satisfies asinh(x) ~ sign(x) log|2x| for large |x|. Other approaches use smooth log-like approximations that are defined for all real inputs and provide a compressive, order-preserving transformation.

Key properties often cited are monotonicity, continuity, and unbounded growth. The derivative often resembles 1/x over substantial ranges, contributing to a similar rate of increase to the natural logarithm in appropriate regions.

Applications include data transformation for variance stabilization, handling zeros or negatives in modeling, and numerical methods that require a log-like scale with improved stability or differentiability.

See also: logarithm, log1p, asinh, Box-Cox transformation, Yeo-Johnson transformation.