I am reading p-adic Numbers, p-adic Analysis, and Zeta-Functions by Neal Koblitz. Please look at page 27. The definition of $f(x)=n^x$ is unambiguously defined when $n$ is $1 \bmod p$. Next, for any $n$ coprime to $p$, he first fixes an $s_0$ and works with $S_{s_0}$. Later he makes the remarks that the function $f$ depends on $s_{0}$ as well as $n$. I didn't quite understand what he means by dependence on $s_0$? Does he mean if you get $n^x$ via two sets $S_{s_0}$ and $S_{s_1}$ for two numbers $s_0$ and $s_1$, then do we get two distinct $p$-adic numbers? Can I see an example where this happens?
My second but related question is, if $a$ is a $p$-adic integer coprime to $p$, and $s$ is an integer in ${0,1,\ldots,p-2}$ and if $v=s/(p-1) \in \mathbb Z_p$, then do we have $a^s=(a^{p-1})^v$ ? (where $(a^{p-1})^v$ is well defined because $a^{p-1} \equiv 1 \bmod p$. )
If the answer to my second question is positive, I suspect all the $p-2$ "branches" of exponentiation corresponding to each $s_0$ are same.
Third question: Why do this fix an $s_0$ business at all? Why not just do this: Given any $x \in \mathbb Z_p$, and $n$ coprime to $p$, just write $x$ as limit of sequence of integers $x_m$ such that $x_m \equiv x_{m-1} \bmod p^m(p-1)$ (which can be achieved by induction) and define $n^x$ as the limit of $n^{x_m}$. I suspect this may fail because I haven't checked if the $x_m$ we get are necessarily non-negative.
Any thoughts will be greatly appreciated. Thanks.