P-adic numbers $ \ \mathbb{Q}_p \ $ and Non-Archimedian analysis:
We know real logarithm from $ \ (0, \infty) \to \mathbb{R} \ $ is a group homomorphism in Archimedian field.
Similarly , we know that the p-adic logarithm expressed by $ \ \log(1+x)=\sum_{n=1}^{\infty} (-1)^{n+1} \frac{x^n}{n} \ $ is also group homomorphism in p-adic field or Non-Archimedian field .
In P-adic field there are both addition and muliplication structure .
My question is-
Can the p-adic logarithm be considered as a ring homomorphism?
Help me with at least a hints also.
First notice that the $p$-adic logarithm is defined only on $1+p \Bbb Z_p$. Indeed, the sequence $a_n := (-1)^{n+1}/n$ satisfies $$\sqrt[n]{|a_n|_p} = p^{v_p(n) / n} \longrightarrow 1, \qquad \text{when}\;\; n \to \infty,$$ so that the series defining $\log(1+x)$ is converging on $p \Bbb Z_p$.
Secondly, the subset $1+p \Bbb Z_p \subset \Bbb Z_p^{\times}$ is actually a multiplicative subgroup. Then it is true that $$\log_p : 1+p \Bbb Z_p \to (\Bbb Q_p,+)$$ is a group homomorphism, i.e. $\log_p(xy) = \log_p(x) + \log_p(y)$. It has no sense to ask whether this is a ring morphism, since $1+p \Bbb Z_p$ has no zero element (so it is not a subring of $\Bbb Z_p$).