7.048. $0.5\left(\lg \left(x^{2}-55 x+90\right)-\lg (x-36)\right)=\lg \sqrt{2}$.
## Solution.

Domain of definition: $\left\{\begin{array}{l}x^{2}-55 x+90>0 \\ x-36>0\end{array}\right.$

From the condition

$$
\begin{aligned}
& 0.5\left(\lg \left(x^{2}-55 x+90\right)-\lg (x-36)\right)=0.5 \lg 2, \lg \frac{x^{2}-55 x+90}{x-36}=\lg 2 \\
& \frac{x^{2}-55 x+90}{x-36}=2
\end{aligned}
$$

We have $x^{2}-57 x+162=0$ for $x \neq 36$. Hence, $x_{1}=54, x_{2}=3$; $x_{2}=3$ does not satisfy the domain of definition.

Answer: 54.