I've been trying to get a more intuitive feel for semi-direct products (in this spirit), and to do so, have been trying to understand what a homomorphism from $\textrm{Af}_1(\mathbb{R}) = \mathbb{R}^+ \rtimes_\phi \mathbb{R}^\times$ (where for each $b \in \mathbb R^\times$ the automorphism $\phi_b$ of $\mathbb R^+$ is given by $a \mapsto ab$) into $\mathbb{R}$ looks like, as opposed to say a homomorphism from the direct product of these groups into $\mathbb{R}$.
Any such homomorphism restricts to the natural log (up to a scalar multiple) along the subgroup $\{0\}\times\mathbb{R}^\times$, and restricts to addition (again up to a scalar multiple) along $\mathbb{R}^+ \times \{1\}$, but I'm less confident how to leverage these observations to describe a general homomorphism when there's the 'twist' than I would be in the direct product case. How would I use this to characterize such homomorphisms?
Edit: In particular, I'd be more interested in continuous homomorphisms, where both groups carry their usual euclidean topologies.