Let $f(x)$ be a non-constant real-analytic function and for real $x$ it satisfies :
$f(2^x) = f(4^x + 2^{x+1} + 2) - f(4^x + 1)$
Before you ask if this simplifies by writing $2^x = y$ note that $2^x$ is never equal to $0$.
I think $f(x)$ is unique upto a multiplication constant $C$.
How to find $f(x)$ ?
What would make an excellent asymptotic to $f(x)$ ?