Phương trình sau đây có bao nhiêu nghiệm\[\left( {{x^2} - 4} \right)\left( {{{\log }_2}x + {{\log }_3}x + {{\log }_4}x + ... + {{\log }_{19}}x - \log _{20}^2x} \right) = 0\]

\[({x^2} - 4)({\log _2}x + {\log _3}x + {\log _4}x + ... + {\log _{19}}x - \log _{20}^2x) = 0( * )\]

Đkxđ: x>0

\(\left( * \right) \Leftrightarrow \left[ {\begin{array}{*{20}{c}}{x = 2(tm)}\\{x = - 2(ktm)}\\{{{\log }_2}x + {{\log }_3}x + {{\log }_4}x + ... + {{\log }_{19}}x - \log _{20}^2x = 0\left( {**} \right)}\end{array}} \right.\)

\[( * * ) \Leftrightarrow \frac + \frac + \frac + ... + \frac - {\left( {\frac} \right)^2}\]

\[( * * ) \Leftrightarrow \frac + \frac + \frac + ... + \frac - {\left( {\frac} \right)^2}\]

\[ \Leftrightarrow logx\left( {\frac{1} + \frac{1} + \frac{1} + ... + \frac{1} - \frac{{lo{g^2}20}}} \right)\]

\[ \Leftrightarrow \left[ {\begin{array}{*{20}{c}}{logx = 0}\\{\frac{1} + \frac{1} + \frac{1} + ... + \frac{1} - \frac{{lo{g^2}20}} = 0}\end{array}} \right.\]

\[ \Leftrightarrow \left[ {\begin{array}{*{20}{c}}{x = 1}\\{\frac{1} + \frac{1} + \frac{1} + ... + \frac{1} = \frac{{lo{g^2}20}}}\end{array}} \right.\]

\[\begin{array}{l} \Leftrightarrow \left[ {\begin{array}{*{20}{c}}{x = 0}\\{\left( {\frac{1} + \frac{1} + \frac{1} + ... + \frac{1}} \right)lo{g^2}20 = \log x}\end{array}} \right.\\ \Leftrightarrow \left[ {\begin{array}{*{20}{c}}{x = 1\left( {tm} \right)}\\{x = {{10}^{\left( {\frac{1} + \frac{1} + \frac{1} + ... + \frac{1}} \right)lo{g^2}20}}\left( {tm} \right)}\end{array}} \right.\end{array}\]

Phương trình (*) có 3 nghiệm.

Đáp án cần chọn là: C