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In this paper, we explore the differences between classical logarithmicfidelity and quantum fidelity. The classical logarithmic fidelity is found tobe always extensive while the quantum one manifests distinct size dependence indifferent phases. Illustrated by the anisotropic Lipkin-Meshkov-Glick model, wefound numerically and analytically that the logarithmic fidelity scales like Nin the symmetry-broken phase and scales like N^0 in the polarized phase. Thesingular behavior around the critical point is also investigated.

The scaling behavior of logarithmic fidelity in quantum phase transition in LMG model