Existence of a continuous mapping.

Question

Do there exists a continuous function $f: \mathbb{R} \to \mathbb{R}_{+}$ such that

$$ f(-\log_x y) = x^{-y},$$ for all $x \in \mathbb{R}_{+}$ and $y \in \mathbb{R}_{+}$ ?

Answer 1

No. If you set $y=1$ with $x>0$ you get $f(0)=1/x$, which is definitely not true for all $x>0$.