Which is larger, $e^\pi$ or $\pi^e$?

Question

I don't know how to approach this.

I tried expanding $e^{\pi}$ using the power series but that was a dead end since I didn't know what to do with it. I tried estimating if $e \log({\pi})$ was greater or $\pi$. And I guess $\pi$ is greater but I don't have a rigorous argument.

An analytical already exists on the site and you can use floor and ceiling functions in that case to conclude $\pi^e<e^{\pi}$ — Archis Welankar, Jan 30 '16 at 14:00
I don't want to solve this using a calculator. I'm guessing e^pi is greater because it's logarithm is probably greater. But, even to compare logarithms I had to resort to sheer numerical calculations. — Saikat, Jan 30 '16 at 14:39

score 2 · Answer 1 · answered Jan 30 '16 at 14:01

2

Hint: Consider

$$f(x) = x^\frac{1}{x}$$ You can check that $f$ attains a maximum at $ x = e$ and so $$e^{1/e}\geq \pi^{1/\pi}$$

Can you continue from here?

answered Jan 30 '16 at 14:01

fosho

6,334

even easier to check $g(x)= \log f(x) = \dfrac{\log x}{x}$ – Henry Jan 30 '16 at 14:04
This is true! Thanks for the observation :) – fosho Jan 30 '16 at 14:04

Which is larger, $e^\pi$ or $\pi^e$?

1 Answers1