Q:

How to prove $\pi^3<3^\pi$ without using explicit value of $\pi$?

How to prove  $\pi^3<3^\pi$ without using explicit value of $\pi$?  In the following link I proposed similar problem and got extremely amazing explanations which uses pure geometrical ideas : How to prove: $2^\frac{3}{2}<\pi$ without writing the explicit values of $\sqrt{2}$ and $\pi$
Here too I am thinking some similar kind of idea (not involving calculus) but not getting any. Can there also be something like that?

A:

Take the logarithm. Divide each side by $3 \pi$. Notice the question becomes: 
$$ \frac{\ln \pi}{\pi} < \frac{ \ln 3}{3}$$
However, the derivative of $ f(x)=\frac{\ln x}{x}$ is $$f'(x)=\frac{1-\ln x}{x^2}$$
From the product rule. So the fucntion $f(x)=\frac{\ln x}{x}$ is decreasing if $x>e$ as $f'(x)<0$. The result follows as. $$\pi >3>e$$As seen here, here and here.

