Question

Find the domain of definition of the following function:
$y={\log }_2(-{\log }_{\frac{1}{2}}(1+\frac{1}{\sqrt[4]{4x}})-1).$

Answer 1

$y={log}_2[-{log}_{\frac{1}{2}}(1+\frac{1}{\sqrt[4]{4x}})-1]$

For $y$ to be real and defined,

$x>0$ and $x\ne 0$

$1+\frac{1}{\sqrt[4]{4x}}>0$, it is always $\forall x>0$

$-{log}_{\frac{1}{2}}(1+\frac{1}{\sqrt[4]{4x}})->0\Rightarrow {log}_2(1+\frac{1}{\sqrt[4]{4x}})=1\Rightarrow 1+\frac{1}{\sqrt[4]{4x}}>2\Rightarrow \frac{1}{\sqrt[4]{4x}}>1\Rightarrow x<1$

Therefore domain of $y$ is $x\in (0,1).$