  The $\log 3$ Theorem, proved by Culler and Shalen, states that every point in
the hyperbolic 3-space is moved a distance at least $\log 3$ by one of the
non-commuting isometries $\xi$ or $\eta$ provided that $\xi$ and $\eta$
generate a torsion-free, discrete group which is not co-compact and contains no
parabolic. This theorem lies in the foundation of many techniques that provide
lower estimates for the volumes of orientable, closed hyperbolic 3-manifolds
whose fundamental group has no 2-generator subgroup of finite index and, as a
consequence, gives insights into the topological properties of these manifolds.
  In this paper, we show that every point in the hyperbolic 3-space is moved a
distance at least $(1/2)\log(5+3\sqrt{2})$ by one of the isometries in
$\{\xi,\eta,\xi\eta\}$ when $\xi$ and $\eta$ satisfy the conditions given in
the $\log 3$ Theorem.
