 Small archimedean value of a large unit

Write $u=\zeta \prod {\epsilon }_i^{b_i}$ in a fixed fundamental-unit basis. The logarithmic embedding is injective on the exponent lattice and lands in the weighted sum-zero hyperplane. Norm equivalence gives a logarithmic coordinate of size at least a constant times $\Vert b{\Vert }_{\infty }$; the weighted sum-zero relation then forces a negative coordinate of comparable size. There are only finitely many torsion choices. This exponential smallness, together with $\alpha u+\beta v=1$ and a linear form in logarithms of algebraic numbers for the nearby value $\beta v$, bounds the unit exponents effectively.