Phương trình sau đây có bao nhiêu nghiệm

* Hướng dẫn giải

$(x^2-4)({\log }_{2}x+{\log }_{3}x+{\log }_{4}x+\mathrm{...}+{\log }_{19}x-{\log }_{20}^{2}x)=0(\ast )$

Đkxđ: x>0

$$\begin{gathered} \left(*\right)\iff \left[\begin{array}{c}x=2\left(tm\right)\\ x=-2\left(ktm\right)\\ {\log }_{2}x+{\log }_{3}x+{\log }_{4}x+\mathrm{...}+{\log }_{19}x-{\log }_{20}^{2}x=0\left(**\right)\end{array}\right. \\ (\ast \ast )\iff \frac{logx}{log2}+\frac{logx}{log3}+\frac{logx}{log4}+\mathrm{...}+\frac{logx}{log19}-{\left(\frac{logx}{log20}\right)}^2 \\ (\ast \ast )\iff \frac{logx}{log2}+\frac{logx}{log3}+\frac{logx}{log4}+\mathrm{...}+\frac{logx}{log19}-{\left(\frac{logx}{log20}\right)}^2 \\ \iff logx\left(\frac{1}{log2}+\frac{1}{log3}+\frac{1}{log4}+\mathrm{...}+\frac{1}{log19}-\frac{logx}{lo{g}^{2}20}\right) \\ \iff [\begin{array}{c}logx\mathrm{=}\mathrm{0}\\ \frac{\mathrm{1}}{log\mathrm{2}}\mathrm{+}\frac{\mathrm{1}}{log\mathrm{3}}\mathrm{+}\frac{\mathrm{1}}{log\mathrm{4}}\mathrm{+}\mathrm{...}\mathrm{+}\frac{\mathrm{1}}{log\mathrm{19}}\mathrm{-}\frac{logx}{lo{g}^{\mathrm{2}}\mathrm{20}}\mathrm{=}\mathrm{0}\end{array} \\ \iff \left[\begin{array}{c}x=1\\ \frac{1}{log2}+\frac{1}{log3}+\frac{1}{log4}+\mathrm{...}+\frac{1}{log19}=\frac{logx}{lo{g}^{2}20}\\ \end{array}\right. \\ \begin{array}{l}\iff \left[\begin{array}{c}x=0\\ \left(\frac{1}{log2}+\frac{1}{log3}+\frac{1}{log4}+\mathrm{...}+\frac{1}{log19}\right)lo{g}^{2}20=\log x\end{array}\right.\\ \iff \left[\begin{array}{c}x=1\left(tm\right)\\ x={10}^{\left(\frac{1}{log2}+\frac{1}{log3}+\frac{1}{log4}+\mathrm{...}+\frac{1}{log19}\right)lo{g}^{2}20}\left(tm\right)\end{array}\right.\end{array} \end{gathered}$$

Phương trình (*) có 3 nghiệm.

Đáp án cần chọn là: C