Question

Find the domain of $f\left(x\right)={\log }_{10}(1-{\log }_{10}(x^2-5x+16\left)\right)$

• A. $(-\infty ,\infty )$
• B. $(-\infty ,0)$
• C. $(0,\infty )$
• D. $(2,3)$

Answer 1

The correct option is D $(2,3)$

For given function to be defined,
(i) $x^2-5x+16>0$ which is true for all $x$. Since discriminant of the quadratic is negative.
(ii) $1-\log (x^2-5x+16)>0\Rightarrow \log (x^2-5x+16)<1\Rightarrow x^2-5x+16<{10}^1$
$\Rightarrow x^2-5x+6<0\Rightarrow (x-2)(x-3)<0\Rightarrow x\in (2,3)$
Considering above two we get required domain $(2,3)$