Question

Solve the equation
$2{\log }_{3}x+{\log }_3(x^2-3)={\log }_{3}0.5+5^{{\log }_5\left({\log }_{3}8\right)}$

Answer 1

This equation can be written as
${\log }_3{x}^2+{\log }_3(x^2-3)={\log }_{3}0.5+{\log }_{3}8$
${\log }_3{x}^2+{\log }_3(x^2-3)={\log }_3\left(4\right)$
This is equivalent to the system
$\{\begin{array}{c}{x}^2>0\\ x^2-3>0\\ x^2(x^2-3)=4\end{array}$
$\Rightarrow \{\begin{array}{c}x<0andx>0\\ x<-\sqrt{3}andx>\sqrt{3}\\ (x^2-4)(x^2+1)=0\end{array}$
$\Rightarrow x^2-4=0\therefore x=\pm 2,butx>0$
consequently $x=2$ is only root of the given equation.