For three positive integers \(p, q, r, x^{p q^2}=y^{q r}=z^{p^{2}r}\) and \(r=p q+1\) such that \(3,3 \log _y x, 3 \log …

For three positive integers \(p, q, r, x^{p q^2}=y^{q r}=z^{p^{2}r}\) and \(r=p q+1\) such that \(3,3 \log _y x, 3 \log _z y, 7 \log _x z\) are in A.P. with common difference \(\frac{1}{2}\). Then \(r-p-q\) is equal to

[JEE Main 2023, 24 Jan (Shift 1)]