Generalization of the Logarithm Function and of the Exponential Function with Arbitrary Base

TL;DR

This paper introduces generalized logarithm and exponential functions with an arbitrary base dependent on a parameter q, analyzing their properties and applying the generalized logarithm to entropy calculations.

Contribution

It proposes a new form of logarithm and exponential functions with a variable base, extending their properties and applications to entropy.

Findings

Derived properties of the generalized functions

Applied the generalized logarithm to entropy expression

Provided insights into base-dependent function generalizations

Abstract

The logarithm function and the exponential function are, by nature, base dependent. Thus, in this paper I introduces an arbitrary base in the logarithm and exponential functions, both dependent on $q$, in order to have $\log_a(x;q)$ and $a_q^x$. Some of the properties of these functions had been analyzed. The logarithm function was applied to the entropy which resulted in the $S_q = k[1 - \sum_{i = 1}^Wp_i^{q}]/[1 - e^{1 - q}]$ expression.