The paper presents a refinement of the log 3 theorem in the study of two-generator free Kleinian groups. The log 3 theorem, established by Culler and Shalen, states that every point in hyperbolic 3-space is moved at least log 3 by one of two non-commuting isometries generating a torsion-free, non-co-compact, parabolic-free group. The authors extend this result to show that every point in hyperbolic 3-space is moved at least log(√(5 + 3√2)) by one of the isometries or their product. This improvement provides a stronger lower bound for the hyperbolic displacements under the group's isometries. The paper explores the implications of this result for the volumes of orientable, closed hyperbolic 3-manifolds with specific topological properties. It builds on the foundational work of Culler, Shalen, and others, who have established lower bounds for the volumes of such manifolds based on their topological and geometric properties. The authors use techniques involving conformal densities, Patterson densities, and the analysis of displacement functions to derive their results. The main result is proven by analyzing the decomposition of the group into subsets of reduced words and applying the properties of conformal densities and the log 3 theorem. The paper also introduces a new decomposition of the group that includes the isometries ξ, η, and ξη, and uses this to derive a lower bound for the displacements under these isometries. The authors show that the infimum of the maximum of certain displacement functions over a simplex provides the lower bound log(√(5 + 3√2)). This result is used to improve the lower bounds for the volumes of hyperbolic 3-manifolds, demonstrating the significance of the log 3 theorem in understanding the geometry and topology of these manifolds. The paper concludes with a detailed analysis of the decomposition of the group and the implications of the results for the study of hyperbolic 3-manifolds.The paper presents a refinement of the log 3 theorem in the study of two-generator free Kleinian groups. The log 3 theorem, established by Culler and Shalen, states that every point in hyperbolic 3-space is moved at least log 3 by one of two non-commuting isometries generating a torsion-free, non-co-compact, parabolic-free group. The authors extend this result to show that every point in hyperbolic 3-space is moved at least log(√(5 + 3√2)) by one of the isometries or their product. This improvement provides a stronger lower bound for the hyperbolic displacements under the group's isometries. The paper explores the implications of this result for the volumes of orientable, closed hyperbolic 3-manifolds with specific topological properties. It builds on the foundational work of Culler, Shalen, and others, who have established lower bounds for the volumes of such manifolds based on their topological and geometric properties. The authors use techniques involving conformal densities, Patterson densities, and the analysis of displacement functions to derive their results. The main result is proven by analyzing the decomposition of the group into subsets of reduced words and applying the properties of conformal densities and the log 3 theorem. The paper also introduces a new decomposition of the group that includes the isometries ξ, η, and ξη, and uses this to derive a lower bound for the displacements under these isometries. The authors show that the infimum of the maximum of certain displacement functions over a simplex provides the lower bound log(√(5 + 3√2)). This result is used to improve the lower bounds for the volumes of hyperbolic 3-manifolds, demonstrating the significance of the log 3 theorem in understanding the geometry and topology of these manifolds. The paper concludes with a detailed analysis of the decomposition of the group and the implications of the results for the study of hyperbolic 3-manifolds.