Giải các bất phương trình :
$\begin{array}{l}
1)\,\,\,{\log _2}\left( {\sqrt {{x^2} - 4x}  + 3} \right) > {\log _{\frac{1}{2}}}\frac{2}{{\sqrt {{x^2} - 4x}  + \sqrt {x + 1}  + 1}} + 1\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)\\
2)\,\,\,{\log _{\frac{1}{3}}}\left( {\sqrt {9x - {x^2}}  + 3} \right) > {\log _3}\frac{{27}}{{\sqrt {9x - {x^2}}  + \sqrt {5 - {x^2}} }} - 3\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(2)
\end{array}$
$1)$ 
Điều kiện: $\left\{ \begin{array}{l} x^2-4x\geq 0\\ x+1\geq 0 \end{array} \right.\Rightarrow \left[\begin{array}{l} x\geq 4\\ -1\leq x\leq 0 \end{array} \right.$
Ta có ${\log _{\frac{1}{a}}}\frac{1}{N} = {\log _a}N$với $\left\{ \begin{array}{l}
a > 0\\
a \ne 1\\
N > 0
\end{array} \right.(*)$
Do đó vế phải của $(1)$ trở thành:${\log _2}\left( {\sqrt {{x^2} - 4x}  + \sqrt {x + 1}  + 1} \right)$
Từ đó:
$\begin{array}{l}
(1) \Leftrightarrow \sqrt {{x^2} - 4x}  + 3 > \sqrt {{x^2} - 4x}  + \sqrt {x + 1}  + 1\\
      \Leftrightarrow \left\{ \begin{array}{l}
\left[ \begin{array}{l}
-1\le x \le 0\\
x \ge 4
\end{array} \right.\\
\sqrt {x + 1}  < 2
\end{array} \right.\\
      \Leftrightarrow \left\{ \begin{array}{l} -1\le x\leq 0\vee x\geq 4\\ x<3 \end{array} \right.
\end{array}\Rightarrow - 1 \le x \le 0$.
Vậy BPT đã cho có nghiệm : $$ - 1 \le x \le 0$$
$2)$
Điều kiện: $\left\{ \begin{array}{l} 9x-x^2\geq 0\\ 5-x^2\geq 0 \end{array} \right.\Leftrightarrow \left\{ \begin{array}{l} 0\le x\le 9\\ -\sqrt5\le x\le \sqrt5 \end{array} \right.\Rightarrow 0\le x\le \sqrt5.$
Áp dụng (*) suy ra vế phải của (2) trở thành
$\log_\frac{1}{3}\frac{\sqrt{9x-x^2}+\sqrt{5-x^2}}{27}-3$
$=\log_\frac{1}{3}(\sqrt{9x-x^2}+\sqrt{5-x^2})$
Vậy BPT đã cho tương đương
$\log_\frac{1}{3}(\sqrt{9x-x^2}+3)>\log_\frac{1}{3}(\sqrt{9x-x^2}+\sqrt{5-x^2})$
$\Leftrightarrow \sqrt{9x-x^2}+3<\sqrt{9x-x^2}+\sqrt{5-x^2}$
$\Leftrightarrow \sqrt{5-x^2}>3$
$\Leftrightarrow 5-x^2>9$
$\Leftrightarrow x^2<-4$(vô lý).
Vậy BPT đã cho vô nghiệm.

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