Answer:
ln(e)/e > ln(π)/π ⇒ e^π > π^e
Explanation:
You want a proof that e^π > π^e.
Comparison function
Consider the function f(x) = ln(x)/x. Its derivative is ...
f'(x) = (1 -ln(x))/x^2
The derivative is zero where ln(x) = 1, or x = e. It is negative for values of x > e, so f(x) has a maximum at x = e and a negative slope for x > e. This means f(x) > f(y) for e ≤ x < y.
Comparison of interest
e^π > π^e
Taking natural logs:
ln(e^π) > ln(π^e)
π·ln(e) > e·ln(π)
ln(e)/e > ln(π)/π . . . . . . . divide by π·e
Based on the above, since e < π, f(e) > f(π), so this comparison statement is true.
ln(e)/e > ln(π)/π ⇒ e^π > π^e . . . . QED