## Task B-3.6.

Determine all values of the parameter $m \in \mathbb{R}$ such that the equation

$$
\log _{x+m}\left(x^{3}-9 x+8\right) \cdot \log _{x-1}(x+m)=3
$$

has a unique solution.
## Solution.

Let's set the conditions of the problem:

$$
\begin{aligned}
& x+m>0, x+m \neq 1 \\
& x-1>0, x-1 \neq 1 \\
& x^{3}-9 x+8>0
\end{aligned}
$$

The given equation can be written in the form

$$
\frac{\log \left(x^{3}-9 x+8\right)}{\log (x+m)} \cdot \frac{\log (x+m)}{\log (x-1)}=3
$$

Furthermore, it holds that

$$
\begin{aligned}
& \log \left(x^{3}-9 x+8\right)=3 \cdot \log (x-1) \\
& \log \left(x^{3}-9 x+8\right)=\log (x-1)^{3} \\
& x^{3}-9 x+8=x^{3}-3 x^{2}+3 x-1 \\
& 3 x^{2}-12 x+9=0 \\
& x^{2}-4 x+3=0
\end{aligned}
$$

The solutions to the last equation are $x=3$ and $x=1$.

1 point

The solution $x=1$ does not satisfy the second and third conditions of the problem, so it cannot be a solution.

1 point

The solution $x=3$ satisfies the second and third conditions, so for it to be the only solution, it must hold that $3+m>0$ and $3+m \neq 1$,