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We prove global existence and uniqueness of strong solutions to the logarithmic porous medium type equation with fractional diffusion(1)∂tu+(-delta)1/2log(1+u)=0, posed for x∈R, with nonnegative initial data in some function space of LlogL type. The solutions are shown to become bounded and C∞ smooth in (x, t) for all positive times. We also reformulate this equation as a transport equation with nonlocal velocity and critical viscosity, a topic of current relevance. Interesting functional inequalities are involved.