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Let x, y, and z be positive real numbers that satisfy

 

\(2 \log_x (2y) = 2 \log_{2x} (4z) = \log_{2x^4} (8yz) \neq 0.\)


The value of \(xy^5 z\) can be expressed in the form \(\frac{1}{2^{p/q}}\), where p and q are relatively prime positive integers. Find p+q.

 Apr 18, 2021
 #1
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$xy^5 z = \frac{1}{2^{16/3}}$, so $p + q = 19$.

 Apr 18, 2021

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