Question

The domain of the function
$f\left(x\right)=log_{4}\left(log_{5} \left(log_{3}\left(18x-x^{2}-77\right)\right)\right)\ldots$ is

• A. $x \in (4, 5)$
• B. $x \in (0, 10)$
• C. $x \in (8, 10)$
• D. $x \in (8, 10]$

Answer 1

Answer: (C)

Since $log\,x$ is defined $\forall\,x > 0$
$f\left(x\right)=log_{4}\left(log_{5} \left(log_{3}\left(18x-x^{2}-77\right)\right)\right)$
$\Rightarrow log_{5}\left(log_{3}\left(18x-x^{2}-77\right)\right) > 0$
$\Rightarrow log_{3}\left(18x-x^{2}-77\right) > 5^{\circ}$
$\Rightarrow log_{3}\left(18x-x^{2}-77\right) > 1$
$\Rightarrow 18x-x^{2} -77 > 3^{1}$
$\Rightarrow \left(x-8\right)\left(x-10\right) < 0$
$\Rightarrow 8 < x < 10$
$\therefore x \in\left(8, 10\right)$