Which is greater $e^{\pi}$ or $\pi^e$?

Question

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Proof that at most one of $e\pi$ and $e+\pi$ can be rational

after solving this one one I was thinking whether $e^\pi$ is greater or $\pi^e$ ? On calculating exact values upto 7 digits i got these values

$e^{\pi} =23.1406926$................and

$\pi^e =22.4591577$...............

So practically I proved that $e^\pi$ is greater than $\pi^e$. But without finding the values If we want to find it , I want to know how to find it .... Suppose there are few type of sim like which one is greater from $17^{51}$ or $51^{17}$ , so we use Binomial Theorem , but here I didn't got any idea how to prove it .

Answer 1

HINT:

$$e^\pi<=>\pi^e\iff e^{\frac1e}<=>\pi^{\frac1\pi}$$

Now find the extreme value(s) of $\displaystyle x^{\frac1x}$

Answer 2

your inequality is equivalent to $\frac{\pi}{\ln (\pi)}>e$ use the function $f(x)=\frac{x}{\ln(x)}$

Answer 3

HINT :

Use $log$, which is increasing on $\mathbb{R^+}$ (then $log(a) > log (b) \implies a>b$ for instance).