Behavior and Solutions of the Transcendental Equation x^p = p^x via the Lambert W Function

Abstract

• The main objective of this work is to study the transcendental equation \( x^p = p^x \), with \( p \in \mathbb{N} \) and \( p \neq 1 \), using the Lambert \( \mathcal{W} \) function. The equation can be rewritten in a form solvable by \( \mathcal{W} \), enabling an explicit determination of its real solutions. We analyze the behavior of these solutions using differential calculus. The number of real solutions depends on the parity of \( p \): there are three solutions for even \( p \) and two for odd \( p \). This work extends prior results (e.g., \cite{Bastos}) by generalizing the analysis beyond prime \( p \). Additionally, we examine the function \( -p^{-1} \ln p \) to understand the asymptotic behavior of the solutions. Finally, we propose a method for obtaining parametrized solutions through two auxiliary equations, which facilitates their computation and provides further insight into the structure of the solutions.