Are $(\mathbb{R} - 0, \times)$ and $(\mathbb{R}, +)$ the same group? What is its name? I'm trying to justify to a friend why it's "not a coincidence" that $a^ba^c = a^{b+c}$, and I want to argue that it's because the structure of $\mathbb{R}$ under addition is exactly the same as the structure of $\mathbb{R} - \{0\}$ under multiplication.
Is "group" the right term to use here? It seems like that to me, since we have closure, inverses, associativity, and an identity. But does a more specific term apply? Am I indeed correct that the groups are the same? Is there a name for this group, and/or examples of it in other contexts?

The groups are not the same: $-1\times -1 = 1$ and $1$ is the identity of $(\mathbb{R}-\{0\},\times)$ but there's no $a\neq 0 \in \mathbb{R}$ such that $a+a = 0$, which is the identity of $(\mathbb{R},+)$.
However, let $\mathbb{R}_+$ denote the positive reals.
Then $(\mathbb{R}_+,\times)$ is isomorphic (as group) to $(\mathbb{R},+)$.
To give an isomorphism, use the exponential map $\operatorname{Exp}: \mathbb{R} \rightarrow \mathbb{R}_+$, which is a bijection and a homomorphism because $e^{a+b} = e^a \times e^b$, as you mention.
