Question

The domain of definition of $f\left(x\right)=\sqrt{\log \left(\log x\right)-\log (4-\log x)-\log 3}$ is

Answer 1

Given, $f\left(x\right)=\sqrt{\log \left(\log x\right)-\log (4-\log x)-\log 3}$
Thus $\log \left(\log x\right)-\log (4-\log x)-\log 2⩾0$
$\Rightarrow \log \left(\frac{\left(\log x\right)(4-\log x)}{3}\right)⩾0$
$\Rightarrow \frac{\log x(4-\log x)}{3}⩾1$
Put $\log x=t$
Therefore, $4t-t^2-2⩾0$
$\Rightarrow t^2-4t+3\le 0$
$\Rightarrow tϵ[1,3]$
$\Rightarrow \log xϵ[1,3]$
Thus $xϵ[{10}^3,{10}^4]$