  For the logarithmically singular parabolic equation \[ u_t-\Delta\ln
u=0\qquad\text{weakly in}\ \ E\times(0,T], \] we establish a Harnack type
estimate in the $L^1_{loc}$ topology, and we show that the solutions are
locally analytic in the space variables and differentiable in time. The main
assumption is that $\ln u$ possesses a sufficiently high degree of
integrability, namely \begin{equation*} \ln u\in
L^\infty_{loc}\big(0,T;L^p_{loc}(E)\big) \quad\text{for some} p\ge1.
\end{equation*} These two properties are known for solutions of singular porous
medium type equations ($0<m<1$), which formally approximate the logarithmically
singular equation. However, the corresponding estimates deteriorate as $m\to0$.
It is shown that these estimates become stable and carry to the limit as
$m\to0$, provided the indicated sufficiently high order of integrability is in
force. The latter then appears as the discriminating assumption between
solutions of parabolic equations with power-like singularities and logarithmic
singularities to insure such solutions to be regular.
