Why $(1+1/2^p)^q = (1+1/2^q)^p$ implies $(p=q)$?

Question

How can I prove that:

$(1+1/2^p)^q = (1+1/2^q)^p$

(real $p\leq q$) implies $p=q$ ?

Seems quite simple, but I don't understand where to start from... Thanx!

Answer 1

It suffices to consider two cases:

$1)$ If $0 < x < y$, then $1 + 2^{-y} < 1 + 2^{-x} => (1 + 2^{-y})^x < (1 + 2^{-x})^x < (1 + 2^{-x})^y$. Contradiction. So $x = y$. In this case take $x = p$, and $y = q$ in the question. Then $p = q$.

$2)$ If $x < 0 < y$, then LHS $> 1 >$ RHS. Done.