## Problem 1

Solve the equation in the set $\mathbb{R}$: $\log _{3}\left(8+2 x-x^{2}\right)=2^{x-1}+2^{1-x}$.
## Problem 1

![](https://cdn.mathpix.com/cropped/2024_06_07_73d71b9fa85a2ce33abfg-2.jpg?height=71&width=1696&top_left_y=664&top_left_x=239)

For $x \in(-2,4) \Rightarrow-\left(x^{2}+2 x+8\right) \in(0,9] \Rightarrow \log _{3}\left(-x^{2}+2 x+8\right) \leq \log _{3} 9=2 \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots . \ldots \mathbf{m}$

![](https://cdn.mathpix.com/cropped/2024_06_07_73d71b9fa85a2ce33abfg-2.jpg?height=78&width=1696&top_left_y=850&top_left_x=239)

So equality can only hold for $\log _{3}\left(-x^{2}+2 x+8\right)=2^{x-1}+2^{1-x}=2 \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots$

The solution is $\mathrm{x}=1$

$1 p$