I am trying to prove that these 2 groups are homomorphic first, since I know that isomorphic groups are isomorphic if they are homomorphic and bijective. My problem is that I don't know which operation to use when proving homomorphism.
$\ln(xy) = \ln(x) + \ln(y)$ shows that the natural logarithm is a homomorphism from the multiplicative group of reals to the additive group. You also need to check that it is one-to-one and onto, to conclude that it is an isomorphism.
As for how you would come up with this operation, I'm not sure that everyone would necessarily come up with this answer on their own. It may be something that was covered in lecture.