Question

$f\left(x\right)={\log }_{100x}\left(\frac{2{\log }_{10}x+1}{-x}\right)$ exists if $x\in$

• A. $(0,{10}^{-2})\cup ({10}^{-2},{10}^{-1/2})$
• B. $[{10}^{-2},{10}^{-1/2})$
• C. $({10}^{-2},{10}^{-1/2})$
• D. $(0,{10}^{-2})$

Answer 1

The correct option is A $(0,{10}^{-2})\cup ({10}^{-2},{10}^{-1/2})$

$f\left(x\right)={\log }_{100x}\left(\frac{2{\log }_{10}x+1}{-x}\right)$
For domain
Firstly $x>0\dots \left(1\right)$ and $100x\ne 1\Rightarrow x\ne \frac{1}{100}\dots \left(2\right)$
and
$\frac{2{\log }_{10}x+1}{-x}>0$
$\Rightarrow \frac{2{\log }_{10}x+1}{x}<0$
$\Rightarrow 2{\log }_{10}x+1<0(\because x>0)$
$\Rightarrow {\log }_{10}x<\frac{-1}{2}$
$\Rightarrow x<{10}^{-1/2}\dots \left(3\right)$
From $\left(1\right)\cap \left(2\right)\cap \left(3\right)$
Hence, domain is $x\in (0,{10}^{-2})\cup ({10}^{-2},{10}^{-1/2})$