An optimization problem has several constraints, and among them I have: $$f(x)\log(1+cg(x,y))\geq z \\ d g(x,y)\leq p \\ 0<p<\infty$$ where $c,d$ are known values.
I combine these constraints as follows: $$f(x)\log(1+\frac{cp}{d})\geq z \\ d g(x,y)\leq p \\ 0<p<\infty$$ where $c,d$ are known positive values.
Can I formulate an equivalent optimization problem in this way?
You replace $g(x,y)$ with $p/d$. You may only do this if the second constraint holds with equality in the optimal solution (of the initial formulation). It depends on the structure of your problem whether that is the case.