Question

Prove that e^pi is bigger than pi^e with out calculating, with proves and evidence
Thank you

Answer 1

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