In this paper, the self-preserving theory of coagulating aerosols is presented in a new way: the logarithmic volume (or mass) distribution of an aerosol undergoing coagulation stays invariant in shape at long times. This is shown for both the free molecular and continuum regime collision frequency functions as well as the constant collision frequency function. In addition, new simple approximate forms are presented for the self-preserving distributions, based on numerical solutions to the discrete coagulation equation.