Kernel of the logarithmic unit embedding

The kernel of $\mathrm{Log}:{\mathcal{O}}_K^{\times }\to {\mathbb{R}}^{r+s}$ is exactly $\mu (K)$. If every conjugate of a unit has modulus one, all its powers lie in a bounded subset of the discrete Minkowski lattice ${\mathcal{O}}_K$, so two powers coincide and the unit is a root of unity.