Question

Identify which of the following statement(s) is(are) correct for the equation ${\log }_{(x^2+2x-3)}(4-{\log }_2(|x-1|+|x-3|))=0$

• A. The equation has exactly two solution
• B. The equation has more than two solutions
• C. The sum of all the solutions of the equation is $6$
• D. The sum of all the solutions of the equation is $4$

Answer 1

The correct option is C The sum of all the solutions of the equation is $6$

${\log }_{(x^2+2x-3)}(4-{\log }_2(|x-1|+|x-3|))=0$
$\log$ function is defined when
$\left(1\right)x^2+2x-3>0\Rightarrow x\in (-\infty ,-3)\cup (1,\infty )$

$\left(2\right)x^2+2x-3\ne 1$
$\Rightarrow x\ne -1\pm \sqrt{5}$

$\left(3\right){\log }_2(|x-1|+|x-3|)=3$
$\Rightarrow |x-1|+|x-3|=8$
$\Rightarrow x=-2,6$
But $x=-2$ is not possible.
$\therefore x=6$