Show (R>0, x) is ismorphic to (R,+)

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.

• Can you think of a function that can turn multiplication into addition? And historically was used for precisely that reason - to simplify multiplication by reducing it to addition? Used in slide rules. – user491874 Nov 12 '17 at 17:08
• The Log function? – animorphlover3 Nov 12 '17 at 17:08
• Correct, and I can see your question has in the meantime been answered with the same answer. – user491874 Nov 12 '17 at 17:09
• What does homomorphic mean? – Derek Holt Nov 12 '17 at 17:26

$\ln(xy) = \ln(x) + \ln(y)$ 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.