Does my proof make sense?

Question

Theorem:
For groups $(\Bbb R,+)$ and $(\Bbb R,*)$ (both only dealing with positive integers) there is a function $\phi$ that turns $(\Bbb R,+)\to(\Bbb R,*)$ and vice versa.

Proof:
Assume $(\Bbb R,+)\to(\Bbb R,*)$. So there is a function where elements $x_1,x_2$ going from additive operation to multiplicative operation where $x_1+x_2 \mapsto x_1 *x_2$. So there is some $\phi$ where $\phi(X_1*X_2)= \phi(x_1)*\phi(x_2)$ (here * is an operation) Take $\phi=e$, so $e^{x_1+x_2}=e^{x_1}*e^{x_2}$. Now the inverse of $e$ is $\ln$, so $\ln(x_1 *x_2) = \ln(x_1+x_2)$

I know there is a lot missing from the proof or at least it's not concrete by looking at it. I just need a little help in cleaning up the theorm and proof.

Answer 1

"Positive integers" should be "positive reals". Integers are whole numbers. Positive reals are usually denoted $\mathbb R_{>0}$. Your map, properly written, should go $(\mathbb R, +) \to (\mathbb R_{>0}, \times)$.

Also, why are you assuming the outcome you are trying to prove? You're asked to provide a function doing the job; just write it down, as you did: $f(x) = e^x$. No need to assume anything.

Other than that and the absence of $\LaTeX$, it is fine (and quite concrete).