Nonlinear equality constraints are non-convex.
The relaxation exp(x)+exp(y)+exp(z)<=1 is convex and can be entered in CVX. If you solve this relaxed problem, and it turns out that equality holds, then you have a solution to the original (equality-constrained) problem. Otherwise, you don’t.
If the relaxation doesn’t provide a solution to the original problem, use a non-convex solver, such as available under YALMIP.