 p-adic logarithm

The p-adic logarithm is defined by $\log (1+x)={\sum }_{n\ge 1}(-1{)}^{n+1}{x}^n/n$ when $|x{|}_p<1$ in a complete extension of ${\mathbb{Q}}_p$. The terms tend to zero in the p-adic absolute value, giving convergence even though the corresponding real series may behave differently.

On a sufficiently small group of principal units, the logarithm is a group homomorphism into the additive field and is inverse to the p-adic exponential function. For $p=2$, it converges on $1+2{\mathbb{Z}}_2$ but becomes injective only after restricting to $1+4{\mathbb{Z}}_2$, since $\log (-1)=0$. It converts multiplicative congruence questions into additive ones while retaining these domain restrictions.