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

TL;DR

This paper presents three visual proofs of the inequality $b^{e}<e^{b}$ for $e<b$, connecting geometric visualizations with the properties of the constants $e$ and $b$.

Contribution

It introduces three novel geometric visual proofs for the inequality $b^{e}<e^{b}$, expanding understanding beyond the specific case of $ ext{pi}^e < e^{ ext{pi}}$.

Findings

Three distinct geometric visual proofs provided.

Enhanced understanding of the inequality $b^{e}<e^{b}$.

Visual approach applicable to related inequalities.

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 $e<b$.