Solve the equation: $\;$ $\sqrt{\log_9 \left(9x^8\right) \cdot \log_3 \left(3x\right)} = \log_3 x^3$
Given equation: $\;\;$ $\sqrt{\log_9 \left(9x^8\right) \cdot \log_3 \left(3x\right)} = \log_3 x^3$
i.e. $\;$ $\sqrt{\left(\log_9 9 + \log_9 x^8\right) \cdot \left(\log_3 3 + \log_3 x\right)} = 3 \log_3 x$
i.e. $\;$ $\sqrt{\left(1 + 8 \times \dfrac{\log_3 x}{\log_3 9}\right) \cdot \left(1 + \log_3 x\right)} = 3 \log_3 x$
i.e. $\;$ $\sqrt{\left(1 + 8 \times \dfrac{\log_3 x}{\log_3 3^2}\right) \cdot \left(1 + \log_3 x\right)} = 3 \log_3 x$
i.e. $\;$ $\sqrt{\left(1 + 8 \times \dfrac{\log_3 x}{2 \log_3 3}\right) \cdot \left(1 + \log_3 x\right)} = 3 \log_3 x$
i.e. $\;$ $\sqrt{\left(1 + 4 \log_3 x\right) \cdot \left(1 + \log_3 x\right)} = 3 \log_3 x$
i.e. $\;$ $1 + 5 \log_3 x + 4 \left(\log_3 x\right)^2 = 9 \left(\log_3 x\right)^2$
i.e. $\;$ $5 \left(\log_3 x\right)^2 - 5 \log_3 x - 1 = 0$
i.e. $\;$ $\log_3 x = \dfrac{5 \pm \sqrt{25 + 20}}{10} = \dfrac{5 \pm \sqrt{45}}{10} = \dfrac{5 \pm 3 \sqrt{5}}{10}$
$\implies$ $x = 3^{\frac{5 + 3 \sqrt{5}}{10}}$ $\;$ or $\;$ $x = 3^{\frac{5 - 3 \sqrt{5}}{10}}$
i.e. $\;$ $x = 3.619$ $\;$ or $\;$ $x = 0.8289$
When $\;$ $x = 3^{\frac{5 - 3 \sqrt{5}}{10}} = 0.8289$, $\;$ the given equation becomes
$\sqrt{\log_9 \left(9 \times 0.8289^8\right) \cdot \log_3 \left(3 \times 0.8289\right)} = \log_3 \left(0.8289\right)^3$
i.e. $\;$ $\sqrt{\log_9 \left(2.006\right) \cdot \log_3 \left(2.4867\right)} = \log_3 \left(0.5695\right)$
i.e. $\;$ $\sqrt{0.3168 \times 0.8292} = -0.5125$
i.e. $\;$ $0.5125 = -0.5125$ $\;\;$ which is not possible.
$\implies$ $x = 3^{\frac{5 - 3 \sqrt{5}}{10}}$ $\;$ is not a valid solution.
When $\;$ $x = 3^{\frac{5 + 3 \sqrt{5}}{10}} = 3.619$, $\;$ the given equation becomes
$\sqrt{\log_9 \left(9 \times 3.619^8\right) \cdot \log_3 \left(3 \times 3.619\right)} = \log_3 \left(3.619\right)^3$
i.e. $\;$ $\sqrt{\log_9 \left(265820.24\right) \cdot \log_3 \left(10.857\right)} = \log_3 \left(47.3986\right)$
i.e. $\;$ $\sqrt{5.6830 \times 2.1707} = 3.5122$
i.e. $\;$ $3.5122 = 3.5122$
$\implies$ $x = 3^{\frac{5 + 3 \sqrt{5}}{10}}$ $\;$ is a valid solution.
$\therefore \;$ The solution to the given equation is $\;\;$ $x = \left\{3^{\frac{5 + 3 \sqrt{5}}{10}} \right\}$