  We prove an inequality that must be satisfied by displacement of generators
of free Fuchsian groups, which is the two-dimensional version of the $\log
(2k-1)$ Theorem for Kleinian groups due to Anderson-Canary-Culler-Shalen. As
applications, we obtain quantitative results on the geometry of hyperbolic
surfaces such as the two-dimensional Margulis constant and lengths of closed
curves, which improves a result of Buser's.
