Prove that if $2^x,3^x, 5^x, 7^x, 11^x ... $ are all integers then $x$ is an integer as well

Question

How easy is it to prove that if $2^x,3^x, 5^x, 7^x, 11^x ... $ are integers then $x$ is an integer as well? I have read the definition of the exponent functions as given in my calculus text, and the question propped up.

Answer 1

Given $ 2^x $, $ 3^x $, $ 5^x $,... are integers. Now suppose $ x $ is not an integer. Suppose it is rational of the form $ p/q $ where $ gcd(p,q) = 1 $. So $ 2^{p/q} $, $ 3^{p/q} $, ... can't be integer, since base of exponents are all prime.