Which one is bigger? $e^{\pi} $ or $\pi^e$

Question

$e^{\pi}$ or $\pi^e$, Can we find which one is bigger by using calculus?

Thanks.

http://math.stackexchange.com/questions/7892/comparing-pie-and-e-pi — lab bhattacharjee, Nov 21 '13 at 10:27
@messel, a little googling before asking a question should help. I would like to know about efficient searching in mSE — lab bhattacharjee, Nov 21 '13 at 11:56
actually,I did not search in google, but when writing question, the site shows similar words,in there it has not been seen because we use diffrent words to ask same question. And in general,typing math question in google is diffucult. — mesel, Nov 21 '13 at 12:02
@messel, https://www.google.co.in/#q=which+is+bigger+pi%5Ee+or+e%5Epi — lab bhattacharjee, Nov 21 '13 at 12:04

score 6 · Accepted Answer · answered Nov 21 '13 at 10:35

6

$$e^\pi>\pi^e\iff\pi>e\log\pi\iff\frac{\log e}e=\frac1e>\frac{\log\pi}\pi$$

Now look at the function

$$f(x):=\frac{\log x}x\;,\;\;x\ge e\implies f'(x)=\frac{1-\log x}{x^2}\le0\implies f(x)$$

is monotone descending, and thus

$$e<\pi\implies \frac1e=\frac{\log e}e=f(e)>f(\pi)=\frac{\log\pi}\pi$$

answered Nov 21 '13 at 10:35

DonAntonio

1 Answers1