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.
We are interested in the discrete logarithm of $A$ in base $g$.
For this reason we generated hundreds of tuples $p,g,D,n$ and the code below correctly computed $n \mod{p^{D-1}}$.
Comments suggest constraints on $g$ but the implementation works for arbitrary $g$.
Adding pari/gp code due to comments
You can run in it in a browser: https://pari.math.u-bordeaux.fr/gp.html
/* discrete logarith modulo p^(D-1)
https://pari.math.u-bordeaux.fr/gp.html
*/
{
dlog1(p,g,a,D=2)=lift(log(lift(a)+O(p^D))/log(lift(g)+O(p^D)));
}
{
tt()=
D=2;
setrand(1);
p=nextprime(10^8);X0=random(p^D);g=Mod(2,p^D);a=g^X0;
X1=dlog1(p,g,a,D);
print([(X1-X0)%p^(D-1)]);
}
tt()