The paper discusses the log 3 theorem, a fundamental result in the study of Kleinian groups, which states that every point in hyperbolic 3-space $\mathbb{H}^3$ 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 that is not co-compact and contains no parabolic elements. This theorem has implications for the volumes of orientable, closed hyperbolic 3-manifolds with specific fundamental groups. The main result of the paper extends this theorem to a set of isometries $\{\xi, \eta, \xi \eta\}$, showing that every point in $\mathbb{H}^3$ is moved a distance at least $\log \sqrt{5 + 3\sqrt{2}}$. The proof involves decomposing the Kleinian group generated by $\xi$ and $\eta$ into disjoint subsets of reduced words and using conformal densities to analyze the hyperbolic displacements of points under these isometries. The paper also introduces an alternative technique to calculate the lower bound $\log \sqrt{5 + 3\sqrt{2}}$ without referring to the common perpendicular of $\xi$ and $\eta$. This technique involves solving a Lagrange multiplier problem to find the minimum of a function over a simplex, leading to the conclusion that the infimum of the maximum of the displacement functions is indeed $\log \sqrt{5 + 3\sqrt{2}}$. The author concludes by emphasizing the significance of this extension for improving lower bounds on the volumes of hyperbolic 3-manifolds and the topological properties of these manifolds.The paper discusses the log 3 theorem, a fundamental result in the study of Kleinian groups, which states that every point in hyperbolic 3-space $\mathbb{H}^3$ 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 that is not co-compact and contains no parabolic elements. This theorem has implications for the volumes of orientable, closed hyperbolic 3-manifolds with specific fundamental groups. The main result of the paper extends this theorem to a set of isometries $\{\xi, \eta, \xi \eta\}$, showing that every point in $\mathbb{H}^3$ is moved a distance at least $\log \sqrt{5 + 3\sqrt{2}}$. The proof involves decomposing the Kleinian group generated by $\xi$ and $\eta$ into disjoint subsets of reduced words and using conformal densities to analyze the hyperbolic displacements of points under these isometries. The paper also introduces an alternative technique to calculate the lower bound $\log \sqrt{5 + 3\sqrt{2}}$ without referring to the common perpendicular of $\xi$ and $\eta$. This technique involves solving a Lagrange multiplier problem to find the minimum of a function over a simplex, leading to the conclusion that the infimum of the maximum of the displacement functions is indeed $\log \sqrt{5 + 3\sqrt{2}}$. The author concludes by emphasizing the significance of this extension for improving lower bounds on the volumes of hyperbolic 3-manifolds and the topological properties of these manifolds.