for example, -log(1+1/x +1/y) in CVX. I want to solve it, x and y are optimizing variables.
You can easily model
log(1+1/x) <=t
using the trick shown at the third line under entropy
https://docs.mosek.com/cheatsheets/conic.pdf
I think that idea can be extended to
log(1+1/x+1/y) <=t
Two approaches are shown in section 5.2.7 Log-sum-inv of https://docs.mosek.com/modeling-cookbook/expo.html#exponential-cone