p-adic logarithm (${\log }_p$)

The $p$-adic logarithm is the convergent series $\log (1+x)={\sum }_{n\ge 1}(-1{)}^{n+1}{x}^n/n$ on a sufficiently small group of principal units.

The $p$-adic exponential $\exp (x)={\sum }_{n\ge 0}{x}^n/n!$ converges near zero and is locally inverse to the $p$-adic logarithm.