With CVX 2.2 and Mosek 9.x or 10.x as solver, just use log’ for logarithm.
If not using Mosek as solver, install CVXQUAD with its exponential.m replacement and use -rel_entr(1,x) in place of log(x). Or you can use -rel_entr_quad(1,x) in place of log(x). But there is no need to ever use rel_entr_quad, which from a user prespective is identical to rel_entr, but which requires CVXQUAD, or it will not run at all).
If Mosek is used, there is only a need or benefit to use CVXQUAD if its quantum, i.e., matrix level, functions are needed. I.e., quantum_entr(X), quantum_rel_entr(X,Y), trace_logm(X,C), trace_mpower(X,t,C), lieb_ando(X,Y,K,t)
For addtional details, see 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 .