See the examples in Perspective of log det function, and CVX formulations of related log det problems using Quantum Relative Entropy from CVXQUAD, which make use of quantum_rel_entr, which is in CVXQUAD GitHub - hfawzi/cvxquad: Implementation in MATLAB-based CVX of various convex/concave functions of matrices (matrix geometric means, quantum relative entropy, ...) .
You don’t need to use CVXQUAD’s exponential.m replacement in order to use CVXQUAD’s quantum_rel_entr.