$8.491 \log _{0.5 \sin 2 x} \sin x=\frac{1}{2}$.
## Solution.

$\log _{\sin x \cdot \cos x} \sin x=\frac{1}{2} \Leftrightarrow \log _{\sin x \cdot \cos x} \sin x=\log _{\sin x \cdot \cos x} \sqrt{\sin x \cdot \cos x} \Leftrightarrow$

$\Leftrightarrow\left\{\begin{array}{l}\sin x>0, \\ \sin x \cdot \cos x \neq 1, \\ \sin x=\sqrt{\sin x \cdot \cos x}\end{array} \Leftrightarrow\left\{\begin{array}{l}\sin x>0, \\ \sin x \cdot \cos x \neq 1, \\ \sin x=\cos x\end{array} \Leftrightarrow\left\{\begin{array}{l}\sin x \cdot \cos x \neq 1, \\ x=\frac{\pi}{4}+2 \pi k\end{array} \Leftrightarrow x=\frac{\pi}{4}+2 \pi k\right.\right.\right.$.

Answer: $x=\frac{\pi}{4}(8 k+1)$.