This question already has an answer here:
This question is a little bit different from Group of positive rationals under multiplication not isomorphic to group of rationals since I was wondering if logarithmic function could solve this or not, thank you.
Consider two groups $(\mathbb Q^+,\cdot)$ and $(\mathbb Q,+)$, does an isomorphism exist between them?
My attempt: Let $\varphi:\mathbb Q^+\rightarrow\mathbb Q$ be the isomorphic function, then the below statement must hold true for all $a,b\in\mathbb Q^+$: $$\varphi(a\cdot b)=\varphi(a)+\varphi(b)$$ So I guess maybe a logarithmic function would be fine here, since it's bijective, too.
But the problem is I can not show for a specific base like $10$ for example, $\log(\mathbb Q^+)=\mathbb Q$.