Constraint to unconstraint optimization problem by subsitution Given the following convex optimization problem

$\min_{x,p} ||x|| - p$
subject to $p > 0$

Can I change the above to an unconstrained convex optimization problem by substituting $c = \log(p)$ and minimize

$\min_{x,c} ||x|| - \exp(c)$

If this is possible I am interested in literature about this topic, but I couldn't find much.
Edit: This is just a minimal example. The problem in the example is of course unbounded, but I think my full optimization problem adds more complexity than necessary to answer my question.

Assuming there are some unstated constraints (otherwise the problem is just unbounded), I would just say
$$min_{x,p} ||x|| - p - \lambda \log(p)$$
where $\lambda$ acts as scaling factor. This is an old-fashioned approach called SUMT that traces back I guess the '60s. It is a seminal method from Fiacco&McCormick. You solve a sequence of convex problem reducing each time lambda, until the solution does not change. These class of methods are part of the interior-point family. They are usually numerically unstable and not very efficient.
As a practical approach, I would instead solve
$$min_{x,p} ||x|| - p$$ 
$$ p\geq \epsilon$$
for decreasing values of $\epsilon$ until either the constraint is not binding, or the solution does not change that much.
