Isomorphism between groups defined by set builder notation How is it possible to show that for a set $G = \{x + y\sqrt{3}| x, y\in\mathbb{Z}\}$ and a set $H = \{2^i*5^j| i,j\in\mathbb{Z}\}$ that the group $[A, +]$ is isomorphic to $[B, *]$ (let $+$ be standard addition and $*$ be standard multiplication between reals)? In particular I am struggling to find a valid function that holds for the homomorphism property.

Yes, because both groups are isomorphic to $(\mathbb Z \times \mathbb Z, +)$.
This gives an isomorphism $G \to \mathbb Z \times \mathbb Z \to H$:
$$
x + y\sqrt{3} \mapsto (x, y) \mapsto 2^x 5^y
$$
To see that this is a homomorphism, compute:
$$
(x + y\sqrt{3})+(x' + y'\sqrt{3})
=(x + x') + (y+y')\sqrt{3}
\mapsto 2^{x+x'} 5^{y+y'} = \cdots
$$
