Slovenia — National Math Olympiad

Rewrite the equation as
$$\begin{align*} 9 &= \log_2(10x) + \log_4(100x) + \log_8(1000x) - 2\log_{64} x \\ &= \log_2(10x) + \frac{\log(100x)}{\log 4} + \frac{\log(1000x)}{\log 8} - \frac{2\log x}{\log 64} \\ &= \log_2(10x) + \frac{\log(100x)}{2\log 2} + \frac{\log(1000x)}{3\log 2} - \frac{2\log x}{6\log 2} \\ &= \log_2(10x) + \frac{1}{2}\log_2(100x) + \frac{1}{3}\log_2(1000x) - \frac{1}{3}\log_2 x \\ &= \log_2(10x) + \log_2(\sqrt{100x}) + \log_2(\sqrt[3]{1000x}) - \log_2\sqrt[3]{x} \\ &= \log_2\left(\frac{10x \cdot \sqrt{100x} \cdot \sqrt[3]{1000x}}{\sqrt[3]{x}}\right) \\ &= \log_2(1000\sqrt[3]{x^3}). \end{align*}$$
It follows that $1000\sqrt[3]{x^3} = 2^9$ or $x = \left(\frac{2^9}{10^3}\right)^{\frac{2}{3}} = \frac{16}{25}$.