= Solution
For sufficiently large $r$, the series for the <p-adic logarithm> and <p-adic exponential> converge on $1+\pi^r\mathcal O_K$ and $\pi^r\mathcal O_K$ and are inverse homomorphisms. Hence the <principal-unit logarithm> gives
$$
1+\pi^r\mathcal O_K\cong(\pi^r\mathcal O_K,+)\cong(\mathcal O_K,+),
$$
where the last map is division by $\pi^r$.
Now take $K=\mathbb Q_3(\zeta_3)$ and $\pi=\zeta_3-1$. This is a <uniformizer>, the residue field is $\mathbb F_3$, and its Teichmuller units are $\mu_2=\{\pm1\}$. Thus
$$
\mathcal O_K^\times\cong\mu_2\times U_1,
\qquad U_i=1+\pi^i\mathcal O_K.
$$
The image of $\zeta_3$ generates $U_1/U_2$, and $\mu_3\cap U_2=1$, so $U_1=\mu_3\times U_2$. The logarithm and exponential already converge inversely on $U_2$, giving $U_2\cong(\pi^2\mathcal O_K,+)\cong(\mathcal O_K,+)$. Since $\mu_2\times\mu_3\cong\mu_6$, this proves the <unit group of Q3 zeta3> decomposition
$$
\mathcal O_K^\times\cong\mu_6\times\mathcal O_K.
$$
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