Question

The domain of the function $f\left(x\right)={\log }_4\left({\log }_5\right({\log }_3(18x-x^2-77))$ is

• A. $x\in (4,5)$
• B. $x\in (0,10)$
• C. $x\in (8,10)$
• D. None of these

Answer 1

The correct option is C $x\in (8,10)$

Since $\log x$ is defined $\forall x>0$
Here $f\left(x\right)={\log }_4\left({\log }_5\right({\log }_3(18x-x^2-77)\left)\right)$
$\Rightarrow {\log }_5\left({\log }_3\right(18x-x^2-77)>0$
$\Rightarrow {\log }_3(18x-x^2-77)>5^{\circ }=1$
$\Rightarrow 18x-x^2-77>3^1$
$\Rightarrow (x-8)(x-10)<0$
$\Rightarrow 8<x<10$

$\therefore x\in (8,10)$