I am currently studying the following equation:
$p^a(1-p)^b=q^a(1-q)^b$
where $p,q \in (0,1)$, and $a,b \in \mathbb{N}$.
I would like to show that the equation is satisfied if and only if $p=q$.
Is it possible to do this in an exact way? I came across this equation when studying dynamical systems, and I don't have much of a background with these sorts of equations.
(Actually, more precisely, I would like to show that
$\sum_k (p^{a_k}(1-p)^{b_k} - q^{a_k}(1-q)^{b_k}) = 0 \Leftrightarrow p = q$
for $p,q \in (0,1)$ and $a_k,b_k \in \mathbb{N}$.)
Thanks