Representation of p-adic integers using primitive roots

Let $p$ be an odd prime and let $r$ be any positive integer that is a primitive root module $p^2$. Let $X_p = \varprojlim \Bbb{Z} / (p-1)p^n \Bbb{Z}$ be the inverse limit of the rings $\Bbb{Z} / (p-1)p^n \Bbb{Z}$. This is useful because for any unit $u \in \Bbb{Z}/p^n\Bbb{Z}$, there exists $K \in \Bbb{Z} / (p-1)p^{n-1}\Bbb{Z}$ with $r^k \equiv u \pmod{p^n}$ for all $k \equiv K \pmod{(p-1)p^{n-1}}$. So, any element of the p-adic integers: $x \in \Bbb{Z}_p$, you can represent it as $x = p^a r^b$ for some $a \in \mathbb{N}, b\in X_p$. I haven't seen this examined at all. Multiplication of terms in $\Bbb{Z}_p$ would just correlate to addition of in $X_p$. There is no operation in $X_p$ that correctly models addition in $\Bbb{Z}_p$. But, exponentiation in $\Bbb{Z}_p$ closely correlates to multiplication in $X_p$. Has anyone seen any examination of this type of representation for p-adic integers?