The conventional approach to solving the single-scattering lidar equation makes use of the assumption of a power law relation between backscatter and extinction with a fixed exponent and constant of proportionality. An alternative formulation is given herein which assumes the proportionality factor in the power law relationship is itself a function of range or extinction. The resulting lidar equation is solvable as before, and examples are given to show how even an approximate description of deviations from the power law form can yield an improved inversion solution for the extinction. A further generalization is given which includes the effects of a background of Rayleigh scatterers.