Question

Solve:$\sqrt{\frac{{\log }_2\left(x-2\right)}{{\log }_{1/2}\left(3x-1\right)}}=1$

Answer 1

$\sqrt{\frac{{\log }_2(x-2)}{{\log }_{\frac{1}{2}}(3x-1)}}=1$

Squaring both sides we get

$\Rightarrow \frac{{\log }_2(x-2)}{{\log }_{\frac{1}{2}}(3x-1)}=1$

$\Rightarrow -\frac{{\log }_2(x-2)}{{\log }_2(3x-1)}=1$

$\Rightarrow {\log }_{(3x-1)}(x-2)=-1$

$\Rightarrow x-2={(3x-1)}^{-1}$

$\Rightarrow (x-2)=\frac{1}{(3x-1)}$

$\Rightarrow (x-2)(3x-1)=1$

$\Rightarrow 3x^2-x-6x+2-1=0$

$\Rightarrow 3x^2-7x+1=0$ where $a=3,b=-7,c=1$

$\therefore x=\frac{7\pm \sqrt{49-4\times 3\times 1}}{2\times 3}$ using the formula $x=\frac{-b\pm \sqrt{b^2-4ac}}{2a}$

$=\frac{7\pm \sqrt{49-12}}{6}$

$=\frac{7\pm \sqrt{37}}{6}$

$\therefore x=\frac{7+\sqrt{37}}{6},\frac{7-\sqrt{37}}{6}$