I've been searching for something similar to the argument below for about a week now and I just must be missing out on the right key words. Can someone point me in the right direction and/or let me know where I'm going wrong? I haven't worked out the details yet; spending most of my time trying to find the article where someone must've worked through all this before...

Let $p$ be an odd prime. Let $g$ be a primitive root mod $p^n$ for all $n \ge 2$. Then for any $a \in \mathbb{Z}$ relatively prime to $p$, we can define the index $\text{ind}_{(g,n)}(a) = k_n$ where $g^{k_n} \equiv a \mod p^n$. Such $k_n$ are only defined modulo $\phi(p^n) = (p-1)p^{n-1}$. But it also must be true that $k_n \equiv k_{n-1} \mod (p-1)p^{n-2}$. Thus for some $0 \le c_i \le p$ and $0\le c_0 \le p-1$, $$k_n \equiv c_0 + c_1(p-1)+c_2(p-1)p+c_3(p-1)p^2+\cdots+c_{n-1}(p-1)p^{n-2} \mod (p-1)p^{n-1}.$$ Hence $\{k_n\}$ converges $p$-adically to some $k \in \mathbb{Q}_p$. Define $\text{ind}_g(a)=k$. Seems reasonable to believe then that $g^k = a$ in $\mathbb{Q}_p$.

Questions:
 - Initially the domain is integers relatively prime to $p$. It seems like these methods can be extended to $z \in \mathbb{Q}$ with $|z|_p = 1$ (for $z=\frac{a}{b}$, just use $ab^{-1}$ computed mod $p^n$). So perhaps it can be defined on any unit of $\mathbb{Q}_p$?
 - How does this relate (if at all) to the [$p$-adic logarithm that shows up in all my internet searches][1]? It seems that $\text{ind}_g$ is defined on the boundary of the domain of $\log_p$. 

Thanks in advance!

 


  [1]: https://en.wikipedia.org/wiki/P-adic_exponential_function#p-adic_logarithm_function