Why $e^x$ is always greater than $x^e$? I find it very strange that $$ e^x \geq x^e \, \quad \forall x \in \mathbb{R}^+.$$ 
I have scratched my head for a long time, but could not find any logical reason. Can anybody explain what is the reason behind the above inequality? I know this is math community, but I'd appreciate a more intuitive explanation than a technical one. 

Here is a different approach, just for the sake of variety, which could be made more rigorous:
Consider the tangent to the curve $y=\ln x$ at the point $x=e,y=1$ is $$y-1=\frac 1e(x-e)\implies y=\frac xe$$
We know that the curve is concave so lies below the tangent except at the tangent point.
Therefore, $$\ln x\leq\frac xe$$
$$\implies x^e\leq e^x$$
