Find the domain of the following function:
$f\left(x\right)=log_4\left(log_5\left(log_3\left(18x-x^2-77\right)\right)\right)$
My text provides a solution which goes like:
=> $log_5\left(log_3\left(18x-x^2-77\right)\right)>0$
=> $log_5\left(log_3\left(18x-x^2-77\right)\right)>log_51$
=> $log_3\left(18x-x^2-77\right)>1$
=> $log_3\left(18x-x^2-77\right)>log_33$
=> $18x-x^2-77>3$
=> $\left(x-8\right)\left(x-10\right)<0$
=> $8<x<10$ (Answer)
But what about
$log_3\left(18x-x^2-77\right)>0$ and $18x-x^2-77>0$.
Shouldn't we consider the above inequalities?
$$log_3\left(18x-x^2-77\right)>log_33\text{ and }log_33>0 \implies log_3\left(18x-x^2-77\right)>0$$ Also $$18x-x^2-77>3\implies 18x-x^2-77>0$$