  We compute the hadronic light-by-light scattering contributions to the muon
anomalous magnetic moment, $\amulbl$, in chiral perturbation theory that are
enhanced by large logarithms and a factor of $N_C$. They depend on a low-energy
constant entering pseudoscalar meson decay into a charged lepton pair. The
uncertainty introduced by this constant is $\pm 60\times 10^{-11}$, which is
comparable in magnitude to the present uncertainty entering the leading-order
vacuum polarization contributions to the anomalous moment. It may be reduced to
some extent through an improved measurement of the $\pi^0\to e^+ e^-$ branching
ratio. However, the dependence of $\amulbl$ on non-logarithmically enhanced
effects cannot be constrained except through the measurement of the anomalous
moment itself. The extraction of information on new physics would require a
future experimental value for the anomalous moment differing significantly from
the 2001 result reported by the E821 collaboration.
