Question

The domain of $f\left(x\right)={\log }_3\{-{\log }_4\left(\frac{6x-4}{6x+5}\right)\}$ is

• A. $(\frac{2}{3},\infty )$
• B. $(-\frac{5}{6},\infty )$
• C. $(-\infty ,-\frac{5}{6})$
• D. $R-\{-\frac{5}{6}\}$

Answer 1

The correct option is A $(\frac{2}{3},\infty )$

For the function $f\left(x\right)={\log }_3\{-{\log }_4\left(\frac{6x-4}{6x+5}\right)\}$ to be defined ,
$\left(i\right)-{\log }_4\left(\frac{6x-4}{6x+5}\right)>0$
$\Rightarrow \frac{6x-4}{6x+5}<1$
$\Rightarrow \frac{6x-4}{6x+5}-1<0$
$\Rightarrow \frac{-9}{6x+5}<0$
$\Rightarrow x>-\frac{5}{6}\dots \left(1\right)$

$\left(ii\right)\frac{6x-4}{6x+5}>0$
$\Rightarrow x\in (-\infty ,-\frac{5}{6})\cup (\frac{2}{3},\infty )\dots \left(2\right)$

$\therefore$From $\left(1\right)$ and $\left(2\right)$
$x\in (\frac{2}{3},\infty )$