Why does CVX accept log(2^exp(x)) but don't accept log(2^exp(x)-1)

log(2^exp(x)) is convex, and log(2^exp(x)-1) should be convex,too.
I search in community and it seems like the CVX simply do not accept it even it is convex.
log(2^exp(x)) is accepted,
log(2^exp(x)+1) is accepted.
log(2^exp(x)-1) with: Illegal operation: log( {convex} ).
Is there any form to approximate it. or some other way out for this problem?
Thank you for your help.

Please read the documentation and the FAQ.

That said, expressions involving log and exp don’t quite follow the DCP rules anyway. Still, they do have their own rules, and they can be found here.

Thank you,it helps me a lot.