Is $π^3$ greater than $3^π$ or conversely?

Question

Using the calculator, we know that

$$3^π > π^3$$

But, without any calculation, how to prove it?

Answer 1

**Hint:$$

Compare the logs: it amounts to checking $\;\dfrac{\ln\pi}\pi < \dfrac{\ln 3}3$.

So you should study the variations of the function $\;\dfrac{\ln x}x$.