Question

The set of values of $x$ for which ${\log }_2(-{\log }_{1/2}(1+\frac{1}{x^4})-1)$ is defined is

• A. $(0,1)$
• B. $(0,1]$
• C. $[-1,1]$
• D. $(-1,1)-\left\{0\right\}$

Answer 1

The correct option is D $(-1,1)-\left\{0\right\}$

Given
${\log }_2(-{\log }_{1/2}(1+\frac{1}{x^4})-1)$
For the $\log$ to be defined, we get
$-{\log }_{1/2}(1+\frac{1}{x^4})-1>0,x\ne 0\Rightarrow -{\log }_{1/2}(1+\frac{1}{x^4})>1\Rightarrow {\log }_{1/2}(1+\frac{1}{x^4})<-1\Rightarrow 1+\frac{1}{x^4}>2\Rightarrow \frac{1}{x^4}>1\Rightarrow x^4<1\Rightarrow x\in (-1,1)\cdots \left(1\right)$

$1+\frac{1}{x^4}>0\Rightarrow \frac{1}{x^4}>-1$
Which is always correct.

Hence, $x\in (-1,1)-\left\{0\right\}$