This is more or less a footnote to Jyrki's excellent answer. I find the idea of $q$-adic analysis slightly disturbing, so I'm going to switch the variable names and talk about the $p$-adic logarithm.
There is a useful field $\mathbb{C}_p$, the completion of the algebraic closure of $\mathbb{Q}_p$, which is in some sense the natural place to do $p$-adic analysis. This contains all the algebraic extensions of $\mathbb{Q}_p$, obviously.
As Jyrki remarks, the power series
$$ \log(x) = \sum_{n \ge 1}\frac{(-1)^{n+1} (x-1)^n}{n} $$
converges $p$-adically for any $x \in \mathbb{C}_p$ whenever $|x - 1| < 1$. All the finite extensions of $\mathbb{Q}_p$ are closed in the topology of $\mathbb{C}_p$, so if $x$ lives in some finite subextension so does its logarithm.
You can extend the log just a bit further by using the group structure. We want to have $\log(xy) = \log(x) + \log(y)$, so any root of unity in $\mathbb{C}_p$ had better go to zero. Now, every $x \in \mathbb{C}_p \setminus \{0\}$ can be written uniquely in the form $x = p^n y z$ where $n \in \mathbb{Z}$, $y$ is a root of unity of order prime to $p$, and $|z - 1| < 1$. Thus once one decides on what $\log(p)$ should be (a "branch of the logarithm"), one has a uniquely determined logarithm map on $\mathbb{C}_p \setminus \{0\}$.