Probably this is easy, but we would like to see it on paper.

Let $p$ be prime and $D,g,n$ positive integers.

Let $A=g^n \bmod p^D$.

Let $\log(p,a,D)$ be the p-adic logarithm with precision $D$. In pari/gp notation: $\log(p,a,D)=$log(a+O(p^D)). $\log(p,a,D)$ is efficiently computable.

Define $dlog(p,g,A,D)=\frac{\log(p,A,D)}{\log(p,g,D)}$.

Is is true that $dlog(p,g,A,D) \bmod{p^{D-1}} = n \bmod{p^{D-1}}$?

We tested hundreds of large tuples for experimental support.

This might have some cryptography application for discrete logarithms modulo prime powers.