Thread: How can I show that these two semigroups is isomorphic?

I have these two semigroups (ℝ≥0, +) and (ℝ≥1, *) where * is multiplication.

How can I show that these are isomorphic?

I know from wikipedia that:
"Two semigroups S and T are said to be isomorphic if there is a bijection f : S ↔ T with the property that, for any elements a, b in S, f(ab) = f(a)f(b). Isomorphic semigroups have the same structure."

But I have no idea on how to go about proving this. Any help would be highly appreciated.

define the following map:

f(x) = e^x.

prove f is a semi-group homomorphism (it's actually a MONOID homomorphism).

then, prove f is bijective.