Question

The equation ${\log }_3(3-x)-{\log }_3(3+x)={\log }_3(1-x)-{\log }_3(2x+1)$ has

• A. two real solutions
• B. one prime solution
• C. no real solution
• D. none

Answer 1

Answer: (D)

We have ${\log }_3(3-x)-{\log }_3(3+x)={\log }_3(1-x)-{\log }_3(2x+1)$
$\Rightarrow {\log }_3((3-x)(2x+1))={\log }_3((3+x)(1-x))\Rightarrow (3-x)(2x+1)=(1-x)(3+x)$
$\Rightarrow 6x-2x^2+3-x=3+x-3x-x^2\Rightarrow -2x^2+5x+3=-x^2-2x+3\Rightarrow x^2-7x=0$
$\Rightarrow x=0,7$
But, $x=7$ (rejected)
So, $x=0$ is the only solution.