Question

The domain of the function $f\left(x\right)={\log }_2\left({\log }_3\right({\log }_4(x^2-3x+6)\left)\right)$ is

• A. $(-\infty ,2)$
• B. $(-\infty ,1)\cup (2,\infty )$
• C. $[0,\infty )$
• D. $[2,\infty )$

Answer 1

The correct option is B $(-\infty ,1)\cup (2,\infty )$

For the function $f\left(x\right)={\log }_2\left({\log }_3\right({\log }_4(x^2-3x+6)\left)\right)$ to be defined,
$\left(i\right){\log }_3\left({\log }_4\right(x^2-3x+6\left)\right)>0$
$\Rightarrow {\log }_4(x^2-3x+6)>1$
$\Rightarrow x^2-3x+6>4^1$
$\Rightarrow x^2-3x+2>0$
$\Rightarrow (x-1)(x-2)>0$
$\Rightarrow x\in (-\infty ,1)\cup (2,\infty )\dots \left(1\right)$

$\left(ii\right){\log }_4(x^2-3x+6)>0$
$\Rightarrow x^2-3x+6>1$

$\left(iii\right)x^2-3x+6>0$

Since we have already solved for $x^2-3x+6$ to be greater than $4$. So, domain is
$x\in (-\infty ,1)\cup (2,\infty )$