Question

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

Answer 1

Domain for $\log xisx>0$

$Let18x-x^2-77=A,{\log }_3(18x-x^2-77)=B,{\log }_5\left(lo{g}_3\right(18x-x^2-77\left)\right)=C$

so from the question the domain need to satisfy the following conditions $A>0,B>0,C>0$

$A>0$ :

$18x-x^2-77>0$

$xϵ(7,11)$

$B>0$ :

$lo{g}_3\left(A\right)>0$

$A>1$

$18x-x^2-78>0$

$xϵ(9-\sqrt{3},9+\sqrt{3})$

$C>0$ :

${\log }_5\left(B\right)>0$

${\log }_3\left(A\right)>1$

$A>3$

$18x-x^2-80>0$

$xϵ(8,10)$

the domain must satisfy every condition so it must be common of all that $xϵ(8,10)$