= Solution
Choose the real embeddings $\sigma_1,\ldots,\sigma_r$ and one embedding $\tau_j$ from each pair of complex-conjugate embeddings. The <logarithmic embedding of number field units> is
$$
\operatorname{Log}:\mathcal O_K^\times\longrightarrow\mathbb R^{r+s},
$$
$$
\operatorname{Log}(u)=
\bigl(log|\sigma_1(u)|,\ldots,\log|\sigma_r(u)|,
2\log|\tau_1(u)|,\ldots,2\log|\tau_s(u)|\bigr).
$$
It is a group homomorphism from multiplication to addition.
If $u$ lies in its kernel, every conjugate of $u$ has modulus one. Under the Minkowski embedding, every power $u^n$ therefore lies in one fixed bounded subset. Each $u^n$ belongs to $\mathcal O_K$, whose Minkowski image is discrete, so that bounded subset contains only finitely many such lattice points. Hence $u^a=u^b$ for some $0\leq a<b$, and
$$
u^{b-a}=1.
$$
\b[Thus $u$ is a root of unity. Conversely, every root of unity clearly has all conjugates of modulus one and lies in the kernel. This proves the <kernel of the logarithmic unit embedding> is precisely $\mu(K)$.]
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