How to implement modified Log function

Hi, I’m looking for ways to implement these functions in CVX:

y*log( 1 + 1/ x )


log( 1 + y /x )


y*log( 1 + y/x )

where x and y are both optimization variables.
All of these functions are from different optimization problems, so I’m already happy when there is a way to implement one of them.

I already had a look at How to handle convex $x_1\log(1+x_2/x_1)$ in DCP ruleset? but its slighly different I guess.

Thanks for your help!

Are they convex? Doesn’t seem so to me. Certainly not \log(1+y/x).

HI! my log function is
log((p*beta/(y+H))+delta);where the p, beta and H are constant .How can i to implement these functions in CVX.