Cho phương trình \({\log _3}\left( {{x^2} - 3} \ri...

C

- Hướng dẫn giải

Phương pháp giải:

Sử dụng các công thức

\(\begin{array}{l}{\log _{{a^n}}}f\left( x \right) = \frac{1}{n}{\log _a}f\left( x \right)\\{\log _a}f\left( x \right) + {\log _a}g\left( x \right) = {\log _a}\left[ {f\left( x \right)g\left( x \right)} \right]\\\left( {0 < a \ne 1;\,\,f\left( x \right) > 0;\,\,g\left( x \right) > 0} \right)\end{array}\)

Giải chi tiết:

ĐK: \(\left\{ \begin{array}{l}{x^2} - 3 > 0\\x - 3 > 0\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}\left[ \begin{array}{l}x > \sqrt 3 \\x <  - \sqrt 3 \end{array} \right.\\x > 3\end{array} \right. \Leftrightarrow x > 3\)

\(\begin{array}{l}{\log _3}\left( {{x^2} - 3} \right) - 2 = {\log _{\frac{1}{3}}}\left( {x - 3} \right)\\ \Leftrightarrow {\log _3}\left( {{x^2} - 3} \right) + {\log _3}\left( {x - 3} \right) = 2\\ \Leftrightarrow {\log _3}\left[ {\left( {{x^2} - 3} \right)\left( {x - 3} \right)} \right] = 2\\ \Leftrightarrow \left( {{x^2} - 3} \right)\left( {x - 3} \right) = {3^2} = 9\\ \Leftrightarrow {x^3} - 3{x^2} - 3x = 0\\ \Leftrightarrow \left[ \begin{array}{l}x = 0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\left( {ktm} \right)\\x = \frac{{3 + \sqrt {21} }}{2}\,\,\,\,\left( {tm} \right)\\x = \frac{{3 - \sqrt {21} }}{2}\,\,\,\left( {ktm} \right)\end{array} \right. \Rightarrow {x_0} = \frac{{3 + \sqrt {21} }}{2}\\P = 2\left( {3 - \sqrt {21} } \right){x_0} = 2\left( {3 - \sqrt {21} } \right).\frac{{3 + \sqrt {21} }}{2} =  - 12\end{array}\)

Chọn C.