Đề bài

Giải mỗi phương trình sau: a) \({\log _4}\left( {x - 4} \right) =  - 2;\) b) \({\log _3}\left( {{x^2} + 2x} \right) = 1;\) c) \({\log _{25}}\left( {{x^2} - 4} \right) = \frac{1}{2};\) d) \({\log _9}\left[ {{{\left( {2x - 1} \right)}^2}} \right] = 2;\) e) \(\log \left( {{x^2} - 2x} \right) = \log \left( {2x - 3} \right);\) g) \({\log _2}{x^2} + {\log _{\frac{1}{2}}}\left( {2x + 8} \right) = 0.\)

Lời giải chi tiết

a) Điều kiện: \(x > 4.\)\({\log _4}\left( {x - 4} \right) =  - 2 \Leftrightarrow x - 4 = {4^{ - 2}} \Leftrightarrow x = \frac{{65}}{{16}}\) (thỏa mãn).b) Điều kiện: \({x^2} + 2x > 0 \Leftrightarrow \left[ \begin{array}{l}x > 0\\x <  - 2\end{array} \right.\) \({\log _3}\left( {{x^2} + 2x} \right) = 1 \Leftrightarrow {x^2} + 2x = 3 \Leftrightarrow {x^2} + 2x - 3 = 0 \Leftrightarrow \left[ \begin{array}{l}x = 1\\x =  - 3\end{array}  \right)\) (thỏa mãn)c) Điều kiện: \({x^2} - 4 > 0 \Leftrightarrow \left[ \begin{array}{l}x > 2\\x <  - 2\end{array} \right.\)\({\log _{25}}\left( {{x^2} - 4} \right) = \frac{1}{2} \Leftrightarrow {x^2} - 4 = {25^{\frac{1}{2}}} \Leftrightarrow {x^2} - 4 = 5 \Leftrightarrow \left[ \begin{array}{l}x = 3\\x =  - 3\end{array}  \right)\) (thỏa mãn)d) Điều kiện: \({\left( {2x - 1} \right)^2} > 0 \Leftrightarrow x \ne \frac{1}{2}.\)\({\log _9}\left[ {{{\left( {2x - 1} \right)}^2}} \right] = 2 \Leftrightarrow {\left( {2x - 1} \right)^2} = {9^2} \Leftrightarrow \left[ \begin{array}{l}2x - 1 = 9\\2x - 1 =  - 9\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}x = 5\\x =  - 4\end{array}  \right)\) (thỏa mãn) e) \(\log \left( {{x^2} - 2x} \right) = \log \left( {2x - 3} \right) \Leftrightarrow \left\{ \begin{array}{l}{x^2} - 2x = 2x - 3\\2x - 3 > 0\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}{x^2} - 4x + 3 = 0\\x > \frac{3}{2}\end{array} \right.\)\( \Leftrightarrow \left\{ \begin{array}{l}\left[ \begin{array}{l}x = 1\\x = 3\end{array} \right.\\x > \frac{3}{2}\end{array} \right. \Leftrightarrow x = 3.\)g) Điều kiện: \(\left\{ \begin{array}{l}{x^2} > 0\\2x + 8 > 0\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}x >  - 4\\x \ne 0\end{array} \right..\) \(\begin{array}{l}{\log _2}{x^2} + {\log _{\frac{1}{2}}}\left( {2x + 8} \right) = 0 \Leftrightarrow {\log _2}{x^2} - {\log _2}\left( {2x + 8} \right) = 0 \Leftrightarrow {\log _2}\frac{{{x^2}}}{{2x + 8}} = 0\\ \Leftrightarrow \frac{{{x^2}}}{{2x + 8}} = 1 \Leftrightarrow {x^2} = 2x + 8 \Leftrightarrow {x^2} - 2x - 8 = 0 \Leftrightarrow \left[ \begin{array}{l}x = 4\\x =  - 2\end{array} \right.\left( {TM} \right).\end{array}\)