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

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 ?

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 ...
• Thank you ! Yes I though about that but I’ve difficulties to see what is the topology of $\mathbb{Z}$ for $v_p$. And about $\langle . \rangle$, this map is quite obscure for me, it comes from a construction and so I can’t really write the expression of its image… I know that for every $\alpha \in \mathbb{Q}_p^\times$ there is such $\langle \alpha \rangle \in 1 + p\mathbb{Z}_p$ but that’s all… May 4, 2022 at 8:20