  We consider the Cauchy problem for the logarithmically singular surface
quasi-geostrophic (SQG) equation, introduced by Ohkitani, $$\partial_t \theta -
\nabla^\perp \log(10+(-\Delta)^{\frac12})\theta \cdot \nabla \theta = 0 ,$$ and
establish local existence and uniqueness of smooth solutions in the scale of
Sobolev spaces with exponent decreasing with time. Such a decrease of the
Sobolev exponent is necessary, as we have shown in the companion paper that the
problem is strongly ill-posed in any fixed Sobolev spaces. The time dependence
of the Sobolev exponent can be removed when there is a dissipation term
strictly stronger than log. These results improve wellposedness statements by
Chae, Constantin, C\'{o}rdoba, Gancedo, and Wu in \cite{CCCGW}.
  This well-posedness result can be applied to describe the long-time dynamics
of the $\delta$-SQG equations, defined by $$\partial_t \theta + \nabla^\perp
(10+(-\Delta)^{\frac12})^{-\delta}\theta \cdot \nabla \theta = 0,$$ for all
sufficiently small $\delta>0$ depending on the size of the initial data. For
the same range of $\delta$, we establish global well-posedness of smooth
solutions to the logarithmically dissipative counterpart: $$\partial_t \theta +
\nabla^\perp (10+(-\Delta)^{\frac12})^{-\delta}\theta \cdot \nabla \theta +
\log(10+(-\Delta)^{\frac12})\theta = 0.$$
