Transforming to a convex constraint

I have the following constraint:

$$log(1+|x|) \le y,$$

where x,y are real scalars.

This domain is a subset of R^2 and it is not convex.
I think that the function log(1+|(x)|) is considered to be quasiconvex (if it helps in any way).
Is there a way to simply transform the domain such that it becomes convex? e.g. to a cone of the form |z|\le w.

The following illustration might help:

If you can replace your variable y with z=e^y, then you could write this as 1+|x|\leq z. But short of that, no.

I guess replacing the variable like that is impossible since it will transform another constraint which is currently convex to a nonconvex one.

So if a transformation like I want doesn’t exists, is it still possible to solve such an optimization using possibly iterative algorithm using CVX?

i.e., is it possible to exploit the specific shape of this constraint?

Quasiconvex energies can be minimized to their unique minimum by using line search but I’m not sure if this is relevant to my case.

My energy is convex but the inequality constraint is such that on the rhs it is affine and the lhs is quasiconvex functions, right?

Is this a quasiconvex constraint (if there is such a thing)?

No, there is no such thing as a quasiconvex constraint. Quasiconvexity can be handled in the objective, but not in the constraint.