Alienation of the logarithmic and exponential functional equations

Zygfryd Kominek,Justyna Sikorska

doi:10.1007/s00010-015-0362-2

Zygfryd Kominek, Justyna Sikorska

Open Access

https://doi.org/10.1007/s00010-015-0362-2

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Abstract

We study the logarithmic–exponential functional equation and check whether the alienation phenomenon takes place.

Highlights

Let E1 = 0 and E2 = 0 be arbitrary functional the equations
We study the logarithmic–exponential functional equation and check whether the alienation phenomenon takes place
We ask whether or not this equation is equivalent to the system of functional equations E1 = 0 and E2 = 0

Summary

Introduction

The aim of the present paper is to solve the functional equation f (xy) − f (x) − f (y) = g(x + y) − g(x)g(y), which is strictly connected with the problem of alienation of logarithmic and exponential Cauchy functional equations for real functions of a real variable. We solve this equation both in the case where we consider all real variables, and—taking into account the nature of a logarithmic function—for non-zero variables.

Functions defined on the whole real line

Equation on a restricted domain