EXPLORING THE APPLICATIONS OF THE LOGARITHMIC MEAN: A PRIMER ON DIFFERENCE CALCULUS

Abstract

The logarithmic mean, often referred to as the log-mean, and has proven its utility across a wide spectrum of disciplines. In this paper, we explore novel applications and uncover its potential in defining the hyperbolic function. Moreover, we introduce a novel approach, which we term the "difference calculus," for deriving two forms analogous to those produced by differential calculus. Notably, our results using this calculus can yield discrete approximations to those obtained via differential calculus. Our discussions predominantly revolve around economic data, assuming positive and discrete variables unless dealing with differentiability-driven scenarios, where continuity and differentiability are presupposed. Additionally, we primarily employ natural logarithms for simplicity. Consider two positive variables, x0 and x1, representing a base period and a comparison period, respectively. We examine their differences, Δx10 = x1 - x0, and logarithmic differences, Δlogx10 = log(x1/x0) = logx1 - logx0. We denote infinitesimal changes as dx and dlogx, with the assumption of non-zero values for interesting outcomes, even when dealing with finite changes of dependent and independent variables in certain functions.