Example 13. Solve the inequality

$$
\log _{1 / 3}\left(x^{2}-6 x+18\right)-2 \log _{1 / 3}(x-4)<0
$$
Solution. The domain of admissible values of the inequality is determined by the system

$$
\left\{\begin{array}{l}
x^{2}-6 x+18>0 \\
x-4>0
\end{array}\right.
$$

From it, we find the domain of admissible values: the interval $44 \\
x^{2}-6 x+18>(x-4)^{2}
\end{array}\right.
$$

Since

$$
\begin{aligned}
\left\{\begin{array}{l}
x>4, \\
x^{2}-6 x+18>(x-4)^{2}
\end{array}\right. & \Leftrightarrow\left\{\begin{array}{l}
x>4, \\
x^{2}-6 x+18>x^{2}-8 x+16
\end{array} \Leftrightarrow\right. \\
& \Leftrightarrow\left\{\begin{array} { l } 
{ x > 4 , } \\
{ 2 x + 2 > 0 }
\end{array} \Leftrightarrow \left\{\begin{array}{l}
x>4 \\
x>-1
\end{array} \Leftrightarrow x>4\right.\right.
\end{aligned}
$$

then the set of all solutions of the original inequality is the interval $4<x<+\infty$.