Question

If $P$ is product of all the values of $x$ satisfying the equation ${\log }_{2x^2-1}(x^2-\frac{2}{3})-2=-\frac{1}{{\log }_3(2x^2-1)}$, then find $4\left|P\right|$.

Answer 1

$\frac{{\log }_3(x^2-\frac{2}{3})}{{\log }_3(2x^2-1)}=2-\frac{1}{{\log }_3(2x^2-1)}$
$\Rightarrow \frac{{\log }_3(x^2-\frac{2}{3})+1}{{\log }_3(2x^2-1)}=2$
$\Rightarrow \frac{{\log }_3({3x}^2-2)}{{\log }_3(2x^2-1)}=2\Rightarrow {3x}^2-2={({2x}^2-1)}^2$

$[\because \log \frac{a}{b}=\log a-\log b\text{\&}m\log a=\log a^m]$

$\Rightarrow 4x^4-7x^2+3=0$
$\Rightarrow x^2=\frac{3}{4}as{x}^2\ne 1$
$\Rightarrow x=\pm \frac{1}{2}\sqrt{3}$
$\therefore P=-\frac{3}{4}\Rightarrow 4\left|P\right|=3$
Ans: 3