If you check Groupisomorphism - does the operation, under which the Groups are closed, matter?

Question

For example: $G_1=(R,+) \quad G_2=(R_+,x)$ So the first group are the Real Number closed under addition and the second group are the positive Real Numbers closed under multiplication. My Lector said that the Isomorphism was $f(x)=e^x$ because its a bijection and $e^(x+y)=e^xe^y$. However, if the 2 Groups were not closed under addition and multiplication would the Groups still be isomorphic? (Btw: I did my research but I didnt really know what I should search for.)