9.114. For what values of $x$ is the following expression defined

$$
\log _{3}\left(1-\log _{0,5}\left(x^{2}-2 x-2.5\right)\right) ?
$$
## Solution.

We obtain the inequality

$$
\begin{aligned}
& 1-\log _{0.5}\left(x^{2}-2 x-2.5\right)>0, \log _{0.5}\left(x^{2}-2 x-2.5\right)0.5, x^{2}-2 x-3>0,(x+1)(x-3)>0 \Leftrightarrow \\
& \Leftrightarrow(x \in(-\infty ;-1) \cup(3 ;+\infty))
\end{aligned}
$$

Answer: $x \in(-\infty ;-1) \cup(3 ;+\infty)$.