The long standing discrepancy between determinations of the CKM matrix element $V_{ub}$ obtained from exclusive and inclusive semileptonic B decays are at the level of 3 standard deviations. This discrepancy continues to challenge our understanding of the semileptonic decays on both the theoretical and experimental sides. Exclusive semileptonic $B_s$ decays are in principle under good theoretical control and provide complementary information with respect to the B-factories in this sector. The article reports the observation of the decay $B^0_s\to K^- \mu^+ \nu_{\mu}$ using 2fb$^{-1}$ of data recorded in 2012 by the LHCb detector. Using the measurement of the $B^0_s\to K^- \mu^+ \nu_{\mu}$ branching fraction relative to the well known $B^0_s\to D_s^- \mu^+ \nu_{\mu}$ decay, and the most recent theoretical knowledge of the relevant $B_s$ decays form factors, the ratio of CKM matrix elements $V_{ub}/V_{cb}$ is extracted, with a precision competitive with the existing measurements.