prove if $b \geq a$, then $a^{b} \geq b^a$

Question

I found that if b = a - 1, then $a^{b} \leq b^{a}$ and if a = b, then $a^{b} = b^{a}$ for obvious reasons. Now, i'm having a hard time figuring out how to prove that if $b \geq a$, then $a^{b} \geq b^a$. I have a few test cases that work but obviously i need to prove this inductively. Any help? also this works for a $\geq$ 4

Answer 1

Hint

Prove and use that the function $x\mapsto \frac{x}{\ln x}$ is increasing on $[e,+\infty)$