7.3092 \log _{9}^{2} x=\log _{3} x \cdot \log _{3}(\sqrt{2 x+1}-1)
Solution. $\frac{\log _{3}^{2} x}{2}=\log _{3} x \cdot \log _{3}(\sqrt{2 x+1}-1) \Leftrightarrow \log _{3} x \cdot\left(\log _{3} x-2 \log _{3}(\sqrt{2 x+1}-1)\right)=0$

![](https://cdn.mathpix.com/cropped/2024_05_21_f024bf2ff7725246f3bfg-100.jpg?height=157&width=763&top_left_y=1132&top_left_x=113)

Answer: $x_{1}=1, x_{2}=4$.

$7.310\left(1+\log _{x} \frac{4-x}{10}\right) \lg x=\lg \lg 10^{3}-1$.

Solution. $\log _{x} \frac{(4-x) x}{10} \cdot \lg x=\lg 3-1 \Leftrightarrow \log _{x} \frac{(4-x) x}{10}=\frac{\lg \frac{3}{10}}{\lg x} \Leftrightarrow$ $\Leftrightarrow \log _{x} \frac{(4-x) x}{10}=\log _{x} \frac{3}{10} \Leftrightarrow\left\{\begin{array}{l}x>0, \\ x \neq 1, \\ (4-x) x=3\end{array} \Leftrightarrow x=3\right.$.

Answer: $x=3$.