$8.37 y=\sqrt{-\frac{\log _{0.3}(x-1)}{\sqrt{-x^{2}+2 x+8}}}$.

## Group 5



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8.37 The following conditions must be satisfied

$$
\left\{\begin{array}{l}
x-1>0 \\
-x^{2}+2 x+8>0 \\
-\frac{\log _{0.3}(x-1)}{\sqrt{-x^{2}+2 x+8}} \geq 0
\end{array}\right.
$$

From the second inequality, it follows that $\sqrt{-x^{2}+2 x+8}>0$, and thus, in the third inequality, $\log _{0.3}(x-1) \leq 0$. Therefore, we have the system

$$
\left\{\begin{array}{l}
x>1 \\
(x-4)(x+2)<0 \\
\log _{0.3}(x-1) \leq 0
\end{array}\right.
$$

Solving which we get $2 \leq x<4$.

Answer: $[2,4)$.