Đáp án:

\(\begin{array}{l}
1)\quad S = \{2;8\}\\
2)\quad S = \{1;4\}\\
3)\quad S = \left\{\dfrac75;\dfrac52\right\}\\
4)\quad S = \{10;100\}
\end{array}\) 

Giải thích các bước giải:

\(\begin{array}{l}
1)\quad \log_2x + 3\log_x2 = 4\qquad (ĐK:x >0;x \ne 1)\\
\Leftrightarrow \log_2x + \dfrac{3}{\log_2x} = 4\\
\Leftrightarrow \log_2^2x - 4\log_2x + 3 =0\\
\Leftrightarrow (\log_2x - 1)(\log_2x - 3) = 0\\
\Leftrightarrow \left[\begin{array}{l}\log_2x = 1\\\log_2x = 3\end{array}\right.\\
\Leftrightarrow \left[\begin{array}{l}x = 2\\x = 8\end{array}\right.\quad \text{(nhận)}\\
\text{Vậy}\ S = \{2;8\}\\
2)\quad \log_2(4x) -\log_{\tfrac x2}2= 3\qquad (ĐK:x >0;x\ne 2)\\
\Leftrightarrow \log_24 + \log_2x - \dfrac{1}{\log_2\dfrac{x}{2}} = 3\\
\Leftrightarrow 2 + \log_2x - \dfrac{1}{\log_2x - 1} = 3\\
\Leftrightarrow (\log_2x + 2)(\log_2x - 1) - 3(\log_2x - 1) - 1 =0\\
\Leftrightarrow \log_2^2x - 2\log_2x = 0\\
\Leftrightarrow \log_2x(\log_2x - 2) =0\\
\Leftrightarrow \left[\begin{array}{l}\log_2x = 0\\\log_2x = 2\end{array}\right.\\
\Leftrightarrow \left[\begin{array}{l}x = 1\\x = 4\end{array}\right.\quad \text{(nhận)}\\
\text{Vậy}\ S = \{1;4\}\\
3)\quad \log_2^4(2x - 3)^2 + \log_2^2(2x - 3)^3 = 25\quad \left(ĐK: x > \dfrac32\right)\\
\Leftrightarrow 16\log_2^4(2x - 3) + 9\log2^2(2x - 3) - 25 =0\\
\Leftrightarrow \left[\log_2^2(2x-3) - 1\right].\left[16\log_2^2(2x-3) + 25\right]=0\\
\Leftrightarrow \left[\begin{array}{l}\log_2^2(2x-3) = 1\\\log_2^2(2x - 3) = - \dfrac{25}{16}\quad (vn)\end{array}\right.\\
\Leftrightarrow \left[\begin{array}{l}\log_2(2x - 3) = -1\\\log_2(2x - 3) = 1\end{array}\right.\\
\Leftrightarrow \left[\begin{array}{l}2x - 3 = \dfrac12\\2x - 3 = 2\end{array}\right.\\
\Leftrightarrow \left[\begin{array}{l}x = \dfrac74\\x = \dfrac52\end{array}\right.\quad \text{(nhận)}\\
\text{Vậy}\ S = \left\{\dfrac75;\dfrac52\right\}\\
4)\quad \dfrac{1}{4 - \log x} + \dfrac{2}{2 + \log x} = 1\qquad \left(ĐK:x > 0;x\ne \dfrac{1}{100};x \ne 10000\right)\\
\Leftrightarrow \dfrac{2+ \log x + 2(4 - \log x)}{(4-\log x)(2 + \log x)} = 1\\
\Leftrightarrow 10 - \log x = (4-\log x)(2 + \log x)\\
\Leftrightarrow \log^2x - 3\log x + 2 =0\\
\Leftrightarrow (\log x -1)(\log x -2) =0\\
\Leftrightarrow \left[\begin{array}{l}\log x = 1\\\log x = 2\end{array}\right.\\
\Leftrightarrow \left[\begin{array}{l}x = 10\\x = 100\end{array}\right.\quad \text{(nhận)}\\
\text{Vậy}\ S = \{10;100\}
\end{array}\)