Question

Find the domain of the function :

$f\left(x\right)=\sqrt{\log \left(\log x\right)-\log (4-\log x)-\log 3}$

• A. $({10}^3,{10}^4)$
• B. $[{10}^3,{10}^4]$
• C. $[{10}^3,{10}^4)$
• D. $({10}^3,{10}^4]$

Answer 1

The correct option is C $[{10}^3,{10}^4)$

$formulaused$
$\log m+\log n=\log \left(mn\right)$
$\log m-\log n=\log \left(\frac{m}{n}\right)$
$f\left(x\right)=\sqrt{\log \left(\log x\right)-\log (4-\log x)-\log 3}$
$\log \left(\log x\right)-\log (4-\log x)-\log 3\ge 0$
$\Rightarrow \log \left(\log x\right)-[\log (4-\log x)+\log 3]\ge 0$
$\Rightarrow \log \left(\log x\right)-\left[\log (12-3\log x)\right]\ge 0$
$\Rightarrow \log \left[\frac{\log x}{12-3\log x}\right]\ge 0$
$\Rightarrow \frac{\log x}{12-3\log x}\ge {10}^0$
$\Rightarrow \frac{\log x}{12-3\log x}\ge 1$
$case1$
$if12-3\log x>0$
$\log x\ge 12-3\log x$
$\Rightarrow 4\log x\ge 12$
$\Rightarrow \log x\ge 3$
$x\ge {10}^3$
$cas2$
$3\log x<12$
$\Rightarrow \log x<x$
$x<{10}^4$
$Hencedomain:[{10}^3,{10}^4)$