For which $\mathbb{Z}_2$ is the logarithm defined?

Question

For which $\mathbb{Z}_2$ is the logarithm defined?

I have that the log series $\log(1+x)=x-\frac{x^2}{2}+\frac{x^3}{3}-\cdots$ gives a convergent series of p-adic integers whenever $x=pz:z\in\mathbb{Z}_2$

Therefore $\log(x)$ is well-defined provided $x$ is a principal unit (of the form $1+pz:z\in\mathbb{Z}_2$), and $\log{a}+\log{b}=\log{ab}$ provided $a,b$ are principal units.

I'm trying to get an idea what these principal units look like in $\mathbb{Z}_2$.

Since they're of the form $1+2z$ I can't escape the ideas that a) these are therefore the numbers ending in a $1$ and b) that would make them the numbers of the form $\lvert x\rvert_2=1$. This would make them the unit group. But surely the unit group must be distinct from the principal units?

So it seems I've got myself in a tangle somewhere.

Answer 1

The units and principal units are exactly the same in $\mathbb{Z}_2$, so you aren't missing anything. The $2$-adic logarithm series is defined on all units of $\mathbb{Z}_2$. (However, some of the nice properties you would expect of the $2$-adic logarithm, such as being the inverse of the $2$-adic exponential, only work when you restrict its domain to elements of the form $1+4z$ for $z\in\mathbb{Z}_2$.)

Answer 2

The case $p=2$ is a bit different than primes $p>2.$ In general, the complete logarithm of a $p$-adic number $x$ consists of three pieces of data: $[v,n,y]\;$ such that $\;x=p^v z_p^n \exp(y)\;$ where $v\in Z,\;$ $z_p$ is a primitive root where $\;z_p^{p-1}=1$ (or $z_p=-1$ if $p=2$), and $y\in pZ_p$ (or $y\in 4Z_p$ if $p=2$).