Có bao nhiêu số nguyên dương a (a là tham số) để phương trình

* Hướng dẫn giải

Điều kiện $0<x<\sqrt{2}$

$PT\iff \left(a^2+4a+5\right){\log }_3\left(2x-x^2\right)+\left(9a^2-6a+2\right){\log }_{11}\left(\frac{2-x^2}{2}\right)={\log }_3\left(2x-x^2\right)+{\log }_{11}\left(\frac{2-x^2}{2}\right)$

$$\begin{gathered} \iff \left(a^2+4a+4\right){\log }_3\left(2x-x^2\right)+\left(9a^2-6a+1\right){\log }_{11}\left(\frac{2-x^2}{2}\right)=0 \\ \iff {\left(a+2\right)}^2{\log }_3\left(2x-x^2\right)+{\left(3a-1\right)}^2{\log }_{11}\left(\frac{2-x^2}{2}\right)=0 \\ \iff {\left(\frac{3a-1}{a+2}\right)}^2=\frac{{\log }_3\left(2x-x^2\right)}{{\log }_{11}\left(\frac{2}{2-x^2}\right)} \left(*\right) \end{gathered}$$