I want to find all continuous functions $f: \mathbb{R}^+ \rightarrow \mathbb{R}^+$ such that $$\forall s \in \mathbb{R}^+: \forall x \in \mathbb{R}^+: \frac{f(x)}{f(s x)} = \frac{f(1)}{f(s)}$$ I know that scaled power functions $f(x)=b x^a$ for some $a \in \mathbb{R}$ and $b \in \mathbb{R}^+$ have this property. I would like to find a continuous function $f$ that has this property although it is not a scaled power function of prove that no such function exists.
(edited after reading Greg Martin's comment)