[en] The chiral properties of the pseudoscalar mesons are studied numerically on a quenched 20⁴ lattice with the overlap fermion. We elucidate the role of the zero modes in the meson propagators, particularly that of the pseudoscalar meson. The non-perturbative renormalization constant Z_A is determined from the axial Ward identity and the pion decay constants f_π is calculated from which we determine the lattice spacing to be 0.157 fm.. We look for quenched chiral log in the pseudoscalar decay constants and the pseudoscalar masses for both equal and unequal quark masses. We find that, given limited statistics, it is difficult to see quenched logs in pseudoscalar masses with equal or unequal quark masses. Whereas, it is relatively clear to see them in the pseudoscalar matrix element and the ratio of the pseudoscalar matrix element to f_π. The chiral log parameter delta turns out to be consistent with that predicted from quenched chiral perturbative theory