Representation of p-adic integers using primitive roots

Let be an odd prime and let be any positive integer that is a primitive root module . Let be the inverse limit of the rings . This is useful because for any unit , there exists with for all . So, any element of the p-adic integers: , you can represent it as for some . I haven't seen this examined at all. Multiplication of terms in would just correlate to addition of in . There is no operation in that correctly models addition in . But, exponentiation in closely correlates to multiplication in . Has anyone seen any examination of this type of representation for p-adic integers?