Prove $e^c>c^e$ if $c>0$ and $e \neq c$ using graph.

Question

I am on this question where it tells me to show $e^c>c^e$ if $c>0$ and $e \neq c$ using the graph of $\dfrac{(log(x))}{x}$.

Now it is obvious that the graph reaches a maximum at $x=e$ but how do i use this graph to show the above statement??
Any hints or advice to get me on the right track would be much appreciated.

Answer 1

Since the exponential function is increasing, it suffices to show that $\log(e^c) > \log(c^e)$ for $c > 0, c \not= e$. Can you do this?