  In this paper, we prove the existence of classical solutions for time
dependent mean-field games with a logarithmic nonlinearity and subquadratic
Hamiltonians. Because the logarithm is unbounded from below, this nonlinearity
poses substantial mathematical challenges that have not been addressed in the
literature. Our result is proven by recurring to a delicate argument, which
combines Lipschitz regularity for the Hamilton-Jacobi equation with estimates
for the nonlinearity in suitable Lebesgue spaces. Lipschitz estimates follow
from an application of the nonlinear adjoint method. These are then combined
with a priori bounds for solutions of the Fokker-Planck equation and a
concavity argument for the nonlinearity.
