Question

If ${\log }_4{\log }_{1/4}{\log }_{4}x>0,$ then $x$ lies in the interval

• A. $(0,\sqrt{2})$
• B. $(1,4)$
• C. $(1,\sqrt{2})$
• D. $(\frac{1}{2},1)$

Answer 1

The correct option is C $(1,\sqrt{2})$

For the given inequality to be defined,
${\log }_{1/4}{\log }_{4}x>0,{\log }_{4}x>0$ and $x>0$
$\Rightarrow {\log }_{4}x<1,x>1$ and $x>0$
$\Rightarrow x<4$ and $x>1$
$\Rightarrow x\in (1,4)\cdots \left(1\right)$

Now, ${\log }_4{\log }_{1/4}{\log }_{4}x>0$
$\Rightarrow {\log }_{1/4}{\log }_{4}x>1$
$\Rightarrow {\log }_{4}x<\frac{1}{4}[\because {\log }_{a}x\text{is strictly decreasing for}a<1]$
$\Rightarrow x<\sqrt[4]{44}=\sqrt{2}\cdots \left(2\right)$

From $\left(1\right)$ and $\left(2\right),$
$x\in (1,\sqrt{2})$