Charakterisation of Homomorphismen from $\mathbb{R} $ to $\mathbb{T}$

Question

I wonder if there is a characterisation of all continous Homomorphismen from $(\mathbb{R}_+, \cdot)$ to $(\mathbb{T},\cdot)$. With $\mathbb{R}_+=(0, \infty)$ and $ \mathbb{T} $ the unit Circle in $ \mathbb{C}$. And $ \cdot $ the usual multiplication.

Answer 1

Sure.$$\begin{array}{ccc}\mathbb{R}_+&\longrightarrow&\mathbb T\\x&\mapsto&e^{in\log(x)}\text,\end{array}$$with $n\in\mathbb Z$.