Codomain of p-adic logarithm We have the natural map
$$\log: \mathbb{C}^\times \to \mathbb{R}$$
$$z \to \log |z|$$
Is there a p-adic analogue of this?
By this I mean, a map $\log_p: \mathbb{C}_p^\times \to \mathbb{Q}_p$, instead of the usual definition of p-adic logarithm, $\log_p: \mathbb{C}_p^\times \to \mathbb{C}_p$.

I’m sure you’ve noticed that since $|z|_p$ is a real number, any logarithm of it is real. In fact, $-\log_p\bigl(|z|_p\bigr)$ is just what’s usually called the $p$-adic valuation of $z$, always an ordinary integer except when $z=0$. Often denoted $v_p(z)$.
