If you are using Mosek as solver with CVX 2.2., it might not matter whether you explicitly use CVX’s exponential sets or rel_entr, provided the formulations are mathematically equivalent. if you are using CVXQIUAD’s exponential.m replacement, only certain formulations cause that to be used (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 ).
Once CVX 2.2 became available (with Mosek 9.x or 10.x)), use of Mosek became the preferred way of solving problems involving exponential cone functions (log, exp, entr, rel_entr, exponential sets, and some others)… You are looking at some very old posts which predate all options other than CVX’s unreliable Successive Approximation method.