2-adic logarithm ($\log (1+x)$)

The 2-adic logarithm is the power series $\log (1+x)={\sum }_{n\ge 1}(-1{)}^{n+1}{x}^n/n$, convergent for $x\in 2{\mathbb{Z}}_2$. Thus it is defined on the odd 2-adic units $1+2{\mathbb{Z}}_2$.

It is inverse to the 2-adic exponential after restriction to $1+4{\mathbb{Z}}_2$, with image $4{\mathbb{Z}}_2$. On the larger domain, $\log (-1)=0$, so it is not injective there. Indeed each odd unit is uniquely a sign times an element of $1+4{\mathbb{Z}}_2$, and the logarithm forgets the sign. Distinguishing the convergence domain from the inverse-function domain avoids a frequent error.