Question

The value of $x$ for which ${\log }_x(x^2-9x+14)={\log }_x|x-2|+{\log }_x|x-7|$ is

• A. $x\in (-\infty ,2)\cup (7,\infty )$
• B. $x\in (2,7)$
• C. $x\in R$
• D. $x\in (0,1)\cup (1,2)\cup (7,\infty )$

Answer 1

The correct option is D $x\in (0,1)\cup (1,2)\cup (7,\infty )$

$x^2-9x+14>0\Rightarrow (x-2)(x-7)>0\Rightarrow x\in (-\infty ,2)\cup (7,\infty )$

For ${\log }_{a}b$ to be defined,
$a>0\text{and}a\ne 1$
$\therefore x\in (0,1)\cup (1,\infty )$

${\log }_x(x^2-9x+14)={\log }_x|x-2|+{\log }_x|x-7|$
$\Rightarrow {\log }_x(x^2-9x+14)={\log }_x|x^2-9x+14|$
So, the equation is true for all value in the domain.