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arXiv · 1108.6096

Algebraic and transcendental solutions of some exponential equations

Jonathan Sondow·Diego Marques
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Abstract

We study algebraic and transcendental powers of positive real numbers, including solutions of each of the equations $x^x=y$, $x^y=y^x$, $x^x=y^y$, $x^y=y$, and $x^{x^y}=y$. Applications to values of the iterated exponential functions are given. The main tools used are classical theorems of Hermite-Lindemann and Gelfond-Schneider, together with solutions of exponential Diophantine equations.

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Jonathan Sondow, Diego Marques. 2011-09-01. Algebraic and transcendental solutions of some exponential equations. https://arxiv.org/abs/1108.6096

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