It is well-known that $\ln\left(\Gamma(x)\right)=x\ln(x)-x+o(x)$. Now consider a integer $q>1$ and $\theta>1$ a real number. Can one write an asymptotic expansion (at least 2 terms) for $$\ln\left(\prod_{j=0}^{n-1}|x-q^j|\right)\quad(n\to+\infty)$$ with $|x|=q^{n\theta}$ ($x\in\mathbb C$)?
Thanks in advance