This paper is concerned with the existence and uniqueness of the solution for the stochastic fast logarithmic equation with Stratonovich linear multiplicative noise in Rd for d⩾3. It provides an answer to a critical case (morally speaking, corresponding to the porous media operator ΔXm for m=0) left as an open problem in the paper Barbu-Röckner-Russo [8]. We face several technical difficulties related both to the degeneracy properties of the logarithm and to the fact that the problem is treated in an unbounded domain. Firstly, the order in which the approximations are considered is very important and different from previous methods. As a by-product of this choice, leading only to weak convergence of relevant terms, identifying the relevant part of the equation in the domain of the nonlinear operator is more fastidious. Secondly, the energy estimates (see eq. (33)) needed in the last step can only be achieved with a well-chosen Stratonovich-type rectification of the noise.