New features in CVX!
The last release 2.2 of CVX comes with support for the exponential cone in MOSEK. It means that problems involving log, exp, rel_entr, log_det, kl_div and other variants of logarithms and exponentials will be passed directly to a single MOSEK solve without employing a successive approximation method.
CVX 2.2 also comes bundled with the very recent MOSEK 9.1.9.
For an example of a log_det problem with MOSEK output in CVX 2.2 see:
http://web.cvxr.com/cvx/examples/log_exp/html/sparse_covariance_est.html
(Note, though, that the CVX warning about using the successive approximation method is still displayed, but if your log output consists of a single Mosek solve like above then it can be ignored.)