"The proof of (5) only goes through for $n\in\ma..."

The proof of (5) only goes through for $n\in\mathbb{N}$.

You can prove a version of (8) from (5), namely, $f(b)=1\Rightarrow f(b^q)=q$ for $q\in\mathbb{Q}$, but this doesn't pin down $f$ completely, unless you include a continuity condition.