Associated Topics || Dr. Math Home || Search Dr. Math

### e^pi vs. pi^e

```
Date: 03/20/2002 at 14:51:19
From: Rajesh Gunesh
Subject: Number Theory

Which is greater, e^pi or pi^e? I would like to have a simple proof,

Thank you!
```

```
Date: 03/20/2002 at 15:28:22
From: Doctor Paul
Subject: Re: Number Theory

Consider the function f(x) = e^x - x^e. Its derivative is

f'(x) = e^x - e*x^(e-1)
f'(x) = 0 will have at most two solutions since f'(x) = 0 implies

e^x = e*x^(e-1)

and it is easily seen that an exponential function and a power
function have at most two points of intersection. Observe that
f'(1) = 0 and f'(e) = 0, so x = 1 and x = e are the two points of
intersection.

In general, an exponential function will grow to infinity faster than
a power function, so f'(x) >= 0 for x > e. This means that f is an
increasing function when x > e.  Also notice that f(e) = 0. Thus,
since Pi > e and f is increasing when x > e, we have f(Pi) > 0.

Thus f(Pi) = e^Pi - Pi^e > 0. This implies that e^Pi is greater than
Pi^e.

A calculator verifies that e^Pi ~= 23.14 and Pi^e ~= 22.46

some more.

- Doctor Paul, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
High School Functions
High School Number Theory
High School Transcendental Numbers

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search