$7.327 \log _{\sqrt{x}}(x+12)=8 \log _{x+12} x, x$-integer.
## Solution.

$\log _{x}(x+12)=\frac{4}{\log _{x}(x+12)} \Leftrightarrow \log _{x}(x+12)= \pm 2 \Leftrightarrow\left\{\begin{array}{l}x \geq 2, x-\text { integer, } \\ {\left[\begin{array}{l}x+12=x^{2} \\ x+12=\frac{1}{x^{2}}\end{array}\right.}\end{array} \Leftrightarrow\right.$ $\Leftrightarrow\left\{\begin{array}{l}x \geq 2, x-\text { integer, } \\ x^{2}-x-12=0\end{array} \Leftrightarrow x=4\right.$

Answer: $x=4$.

$7.3285^{\lg x}-3^{\lg x}=5,(3) \cdot 3^{0.5 \lg x} \cdot 5^{0.5(\lg x-2)}$.

Solution. Let $u=5^{\frac{1}{2} \lg x}, v=3^{\frac{1}{2} \lg x}$, then $u^{2}-v^{2}=5 \frac{1}{3} \cdot v \cdot \frac{u}{5} \Leftrightarrow$ $\Leftrightarrow 15 u^{2}-16 u v-15 v^{2}=0 \Leftrightarrow u=\frac{8 v \pm \sqrt{64 v^{2}+225 v^{2}}}{15} \Leftrightarrow u=\frac{8 v \pm 15 v}{15} \Leftrightarrow$ $\Leftrightarrow\left[\begin{array}{l}u=-\frac{v}{3}, \\ u=\frac{5}{3} v\end{array} \Leftrightarrow 3 u=5 v \Leftrightarrow 5^{\frac{1}{2} \lg x-1}=3^{\frac{1}{2} \lg x-1} \Leftrightarrow \frac{1}{2} \lg x-1=0 \Leftrightarrow x=100\right.$.

Answer: $x=100$.