Showing isomorphism between two groups (true or false question)

Question

I am asked if the group $\mathbb Q_+ = \{ q \in \mathbb Q\, | \,q >0 \} $ (under multiplication) is isomorphic to $\mathbb Z$ (under addition). How do I do this? I am just confused where to start.

Answer 1

Imagine there was an isomorphism $\phi : \mathbb{Z} \to \mathbb{Q}$. Then just as $1$ generates all of $\mathbb{Z}$, $\phi (1) = \frac{a}{b}$ would generate all of $\mathbb{Q}$, but this is clearly impossible (for example, it does not generate $\frac{a+1}{b}$).