Geometric visualizations of $b^{e}<e^{b}$ when $e<b$

Abstract

In connection to the two fascinating constants $e$ and $\pi$, there are many beautiful visual proofs to the inequality $\pi^{e}<e^{\pi}$. The aim of this classroom capsule is to give three visual proofs to the more general inequality $b^{e}<e^{b}$ where $e<b$.