An interesting property of the number e: Conjecture

Though the JavaSketchpad sketch does not allow you to accurately and precisely change the value of the base a, you should hopefuly have noticed that in the 1st quadrant only when a = 2.71..., the two graphs touch once, and once only. This implies that for base a = e the exponential function is always greater than or equal to the power function; in other words, e^x ≥ x^e, and equality only holds when x = e.
This is a remarkable property of e, which implies that for any value of x ≥ 0 other than e: e^x > x^e. For example, e² > 2^e, e³ > 3^e, e^π > 2^π, and even for e^0.001 > 0.001^e, e⁹⁹⁹ > 999^e, etc.

Challenge: Can you explain why (prove that) the result is true? Check your logical explanation (proof) here.