Experiments on ferromagnetic thin films have measured the dependence of the hysteresis loop area on the amplitude and frequency of the external field, A=A(H0,ω), and approximate agreement with numerical simulations of Ising models has been reported. Here we present numerical and theoretical calculations of A in the low-frequency regime for two values of H0, which bracket a temperature and system-size dependent crossover field. Our previous Monte Carlo studies have shown that the hysteretic response of the kinetic Ising model is qualitatively different for amplitudes above and below this crossover field. Using droplet theory, we derive analytic expressions for the low-frequency asymptotic behavior of the hysteresis loop area. In both field regimes, the loop area exhibits an extremely slow approach to an asymptotic, logarithmic frequency dependence of the form A∝-[ln(H0ω)]-1. Our results are relevant to the interpretation of data from experiments and simulations, on the basis of which power-law exponents for the hysteresis-loop area have been reported. © 1998 American Institute of Physics.