Elementary Proofs of Algebraic Relationships for the Exponential and Logarithm Functions

Abstract

This paper uses elementary algebraic methods to obtain new proofs for theorems on algebraic relationships between the logarithmic and exponential functions. The main result is multivariate version of a special case of the Structure Theorem due to Risch that gives in a very explicit fashion the possible algebraic relationships between the exponential and logarithm functions. In addition there are some more results that give new information about the forms of elementary integrals of elementary functions as well as a new treatment of some algebraic dependence theorems previously discussed by Ostrowski, Kolchin and Ax.