I want to show that $ e^{\pi} > \pi ^{e}$?

I was trying to make some functional relations to verify this but I am not able to do so . Any help or hints will be helpful for me.



marked as duplicate by Micah, Namaste, Belgi, Git Gud, Andrés E. Caicedo Apr 20 '13 at 17:42

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  • 5
    $\begingroup$ I think this was asked here before. $\endgroup$ – Pedro Tamaroff Apr 20 '13 at 17:31
  • $\begingroup$ Take logarithms on both the side and see what you got? $\endgroup$ – srijan Apr 20 '13 at 17:31

Let $a = e^{\pi}$ and $b = \pi ^{e}$.

Taking logarithms we obtain $a > b$ iff $\frac {\log e}{e} > \frac {\log \pi}{\pi}$

Now consider the function $f(x) = \frac{\ln x}{x}$ and check when $f$ is decreasing?


Look at the function $$f(x)=\frac{\log x}x$$

You're looking at $x^y<y^x$, or equivalently $f(x)<f(y)$.


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