No CVX, successively approximates the log and solves a new convex optimization problem on each iteration.
If you use mosek as solver, then its native exponential cone capability will be utilized instead. This should be faster and more reliable than CVX’s successive approximation method.
The next best way (other than mosek) to solve this in CVX is to use CVXQUAD, which uses a 2 by 2 SDP approximation of the log, which CVX converts into an SOCP, and as discussed in CVXQUAD: How to use CVXQUAD's Pade Approximant instead of CVX's unreliable Successive Approximation for GP mode, log, exp, entr, rel_entr, kl_div, log_det, det_rootn, exponential cone. CVXQUAD's Quantum (Matrix) Entropy & Matrix Log related functions