Why $p$-adic logarithm is continuous on $\mathbb{Q}_p^\times$?

Question

Neukirch defined, in Algebraic Number Theory, Neukirch (5.4) p. 136, the $p$-adic logarithm on $1 + p\mathbb{Z}_p$ as $\log(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \dots$ Then, he extends this logarithm to $\mathbb{Q}_p^\times$ by $\log(\alpha)=v_p(\alpha)C + \log(\langle \alpha \rangle)$ where $C$ is carefully chosen, $\alpha=p^{v_p(\alpha)}\omega(\alpha)\langle \alpha \rangle \in \mathbb{Q}_p^\times$, $\omega(\alpha) \in \mu_{p-1}$ and $\langle \alpha \rangle \in 1 + p\mathbb{Z}_p$. He says that this extension is clearly continuous. But I don’t see why… Could you help me ?

Answer 1

Hint: Are the maps $\alpha \mapsto v_p(\alpha)$ and $\alpha \mapsto \langle \alpha \rangle$ continuous on $\mathbb Q_p^*$? Is the power series $log$ continuous on $1+p\mathbb Z_p$? And composition, sum, product of continuous maps are ...