A solution to $f(2x)=\alpha f(x)$ with a boundary condition $f(\beta) = \beta$ is $$ f(x) = \left( \frac{\beta}{\alpha^{\log_2 \beta}} \right) \alpha^{\log_2 x}. $$
Do we know whether or not the solution is unique?
Edit: the function should be real-valued of a real variable, continuous, and monotonic.