  This paper is concerned with the existence and uniqueness of the solution for
the stochastic fast logarithmic equation with Stratonovich multiplicative noise
in $\mathbb{R}^{d}$ for $d\geqslant 3$. It provides an answer to a critical
case (morally speaking, corresponding to the porous media operator $\Delta X^m$
for $m=0$) left as an open problem in the paper Barbu-R\"ockner-Russo (Journal
de Math\'ematiques Pures et Appliqu\'ees,103(4):1024--1052, 2015). 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. Secondly, the energy estimates needed in
the last step can only be achieved with a well-chosen Stratonovich-type
rectification of the noise.
