abstract algebra - How to decide the cardinality of ${text{all group isomorphisms from }(mathbb{R},+)text{ onto }(mathbb{R}^+,cdot)}$?

The additive group of reals $(\mathbb{R},+)$ and the multipilicative group of positive reals $(\mathbb{R}^+,\cdot)$ are isomorphic, and $x \mapsto \exp(x)$ is one isomorphism from $(\mathbb{R},+)$ onto $(\mathbb{R}^+,\cdot)$. This is probably an elementary example of group isomorphism that every beginning learner knows. But I wonder something more.

Problem:
How to decide the cardinality of $\{\text{all group isomorphisms from }(\mathbb{R},+)\text{ onto }(\mathbb{R}^+,\cdot)\}$?

• Every function of the form $x \mapsto c^x$ in which $c \in (0,\infty) \backslash \{1\}$ is one continuous group isomorphism from $(\mathbb{R},+)$ onto $(\mathbb{R}^+,\cdot)$. But is every continuous group isomorphism from $(\mathbb{R},+)$ onto $(\mathbb{R}^+,\cdot)$ is exactly of this form?




• How to consider those group isomorphisms that contains discontinuities?

I have got hinted with something like that the/an answer to this problem depends on whether we have the axiom of choice or NOT. I got people hinted me on the freenode IRC in channel ##math, but that place (IRC) is not a good place to elaborate on things about this problem to a beginning learner. So I am asking more detailed elaboration on things about this problem here.

Thank you in advance for any possible help you may give!🙂🙂🙂