Question 9
Let $x,$ $y,$ and $z$ be positive real numbers that satisfy $$
\begin{aligned} 2\log_{x}(2y) &= 2\log_{2x}(4z) \\ &= \log_{2x^4}(8yz) \ne 0. \end{aligned}
$$ The value of $xy^5z$ can be expressed in the form $\frac{1}{2^{\frac{p}{q}}},$ where $p$ and $q$ are relatively prime positive integers. Find $p + q.$
\begin{aligned} 2\log_{x}(2y) &= 2\log_{2x}(4z) \\ &= \log_{2x^4}(8yz) \ne 0. \end{aligned}
$$ The value of $xy^5z$ can be expressed in the form $\frac{1}{2^{\frac{p}{q}}},$ where $p$ and $q$ are relatively prime positive integers. Find $p + q.$