For $p\neq 2$ it's easy to prove through the log/exp-correspondence that
$$(1+p\mathbb{Z}_p)^{p^k}=1+p^{k+1}\mathbb{Z}_p.$$
This gives an easy way to compute the groups $\mathbb{Q}_p^*/\mathbb{Q}_p^{*n}$ which are useful in a lot of computations involving local class field theory. I was wondering if there's some nice way to express the above equation in the case $p=2$?