Look at he Logistic regression equations under Figure 7.1 on p. 356 of Boyd and Vandenberghe “Convex Optimization” http://web.stanford.edu/~boyd/cvxbook/bv_cvxbook.pdf . And see the code at http://web.cvxr.com/cvx/examples/cvxbook/Ch07_statistical_estim/html/logistics.html .
That should allow you to use log_sum_exp .(Note that 1 = exp(0), as the first exp term being summed inside the log).
As listed 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
log_sum_exp is compatible with invoking CVXQUAD’s Padé Approximant instead of CVX’s Successive Approximation method. You can deal with the exp terms outside the log using either of the reformulations described in the just provided link for exp(cvx_expression) .