+0

0
48
1
+79

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

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