= 2-adic logarithm
{title2=$\log(1+x)$}
The <2-adic logarithm> is the <power series> $\log(1+x)=\sum_{n\ge1}(-1)^{n+1}x^n/n$, convergent for $x\in2\mathbb Z_2$. 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.
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