This paper derives a non-linear diffusion equation discussing two possible applications: the ionic diffusion in glasses and temperature-dependent conductivity in semiconductors. The first equation is a logarithmic diffusion derived formally from the continuity of ion concentration, but the latter is a more phenomenological example. A power-law ansatz with time-dependent parameters maximizes a non-standard entropy and gives a set of coupled motion equations we can solve analytically. These results build the general solution to the non-linear diffusion equation.