While trying to prove that two groups are isomorphic I noticed how similar the problem is to finding a solution to a functional equation and then proving that the function is bijective. For instance in the following problem:
Prove that $\mathbb{R^{+}(\cdot)}$ is isomorphic to $\mathbb{R}(+)$
We need to find a bijective function such that $f(x \cdot y)=f(x)+f(y)$. One such function is $f(x)=\log(x)$ because of the property $\log(x \cdot y) = \log(x)+\log(y)$. Now I've had some experience with functional equations but it's very little.
Is there a way to transform every problem that involves finding an isomorphism between groups to solving a functional equation?
In the example I mentioned I believe it is possible but in many cases there is very little information about the operation defined on the group.