Let $X$ be a smooth quasi-projective variety over $\mathbb C$.
Let us say that $X$ is of log-general type if for some choice of smooth compactification $\bar X$ with normal crossings boundary divisor $D = \bar X \setminus X$, we have that the pair $(\bar X, D)$ is of log-general type. The latter means that $K_{\bar X} + D$ is big.
Is this an intrinsic property? That is, is it independent of the choice of $\bar X$ and $D$?
Also, suppose now that $X$ is only a normal quasi-projective variety. I would like to say $X$ is of log-general type if some desingularization of $X$ is of log-general type. Is this well-defined? That is, is this independent of the chosen desingularization of $X$?