The matter density field exhibits a nearly lognormal probability density distribution after entering into the nonlinear regime. Recently, it has been shown that the shape of the power spectrum of a logarithmically transformed density field is very close to the linear density power spectrum, motivating an analytic study of it. In this paper, we develop cosmological perturbation theory for the power spectrum of this field. Our formalism is developed in the context of renormalized perturbation theory, which helps to regulate the convergence behavior of the perturbation series, and of the Taylor series expansion we use for the logarithmic mapping. This approach allows us to handle the critical issue of density smoothing in a straightforward way. We also compare our perturbative results with simulation measurements.