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#### eddybob123

##### Active member

- Aug 18, 2013

- 76

Without using any computing devices, show which number is larger: $e^\pi$ or $\pi ^e$.

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- #1

- Aug 18, 2013

- 76

Without using any computing devices, show which number is larger: $e^\pi$ or $\pi ^e$.

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- #2

- Aug 18, 2013

- 76

Hint:

Consider the function $y=\frac{x}{\ln(x)}$.

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Typically, you should not expect a solution to be posted so quickly; we ask in our http://mathhelpboards.com/challenge...nswering-challenging-problem-puzzle-3875.html that you give our members about a week to respond. Sometimes you will get a quick response, but sometimes not.So does no one know or no one bothers to post their solution?...

In my case, I have seen this problem before along with its solution, so I felt it would only be fair to leave it for the enjoyment of someone who has not seen it before.

The vast majority of problems posted as challenges here are solved, but most people are not online all the time, and so it may be a while before someone comes along who will solve the problem and post their solution.

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- Aug 18, 2013

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I haven't even had my breakfast yet.So does no one know or no one bothers to post their solution?

\(\displaystyle \frac{dy}{dx}=\frac{\ln x -1}{(\ln x )^2}\)

So \(\displaystyle \frac{dy}{dx}=0 \text{ when } x=e \text{ and } \frac{dy}{dx}>0 \text{ when } x>e\)

\(\displaystyle y=e\) when \(\displaystyle x=e\) and \(\displaystyle y>e\) when \(\displaystyle x>e\).

So when \(\displaystyle x=\pi\) we have \(\displaystyle \frac{\pi}{\ln \pi}>e \Rightarrow \pi>e \ln \pi \Rightarrow e^{\pi}> \pi^ e \)

Actually, I haven't even done breakfast for my children.

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- Aug 18, 2013

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- Mar 31, 2013

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that

x^(1/x) is maximum at x = e so e^(1/e) > π^(1/π)

and hence e^π > π^e after raising both sides to power πe