Axiom pinpointing has attracted extensive interest in Description Logics (DLs) due to its effect of exploring explicable defects in the DL ontology and searching for hidden justifications for logical implication. Balancing the expressive power of DLs and the solving efficiency of reasoners has always been the focus of axiom pinpointing research. This study, from both glass-box and black-box perspectives, proposes a consequence-based method for axiom pinpointing. The glass-box method uses modified consequence-based rules (pinpointing rules) to trace the specific process of inference and introduces the concept of pinpointing formula to establish the correspondence between the label of the Boolean formula and all the minimal axiom sets. The black-box method directly calls the reasoner based on the unmodified consequence-based rules and further uses the Hitting Set Tree (HST) to compute all justifications of logical implication. Finally, a reasoning tool is designed based on the two axiom pinpointing algorithms for expressive DL ontologies. Its feasibility is verified theoretically and experimentally, and its solving efficiency is compared with that of existing axiom pinpointing tools.