How to show $\theta$ is a homomorphism from $(\Bbb C, \times)$ onto $(\Bbb R_+, \times)$.
Let $\theta: \Bbb C^∗ \to \Bbb R_+$ by $\theta (z)=|z|$
What I have so far is:
$$\theta(ab) = |ab|$$
$$\theta(a) + \theta(b) = |a||b| = |ab|$$
does this mean that they are homomorphism?
Let $(G, \circ)$ and $(H, \diamond)$ be groups. A homomorphism is a map $\theta: G \rightarrow H$ that preserves the group operations. I.e. for $g, h \in G$
$$\theta(g \circ h) = \theta(g) \diamond \theta(h).$$
In your case, $G = \mathbb{C}^{*}$ and $H = \mathbb{R}^+$ with the operations of multiplication in both cases.
Let $\theta(z) = \lvert z \rvert$ with $z \in \mathbb{C}^*$. To check if this is a homomorphism, we simply apply the definition. Let $z_1$ and $z_2$ be elements of $\mathbb{C}^*$. Then,
$$\theta(z_1z_2) = \lvert z_1 z_2\rvert = \lvert z_1\rvert \lvert z_2 \rvert = \theta(z_1)\theta(z_2),$$
as was to be shown.