Question

If $(m, n)$ represents the domain of the function defined as $f(x)=\sqrt{\log _{10}\left\{\frac{\log _{10} x}{2\left(3-\log_{10} x\right)}\right\}}$. Find $\frac{n}{m}$

Answer 1

$f(x)$ is defined if $\log _{10} \frac{\log _{10} x}{23-\log _{10} x} \geq 0$ and $\frac{\log _{10} x}{23-\log _{10} x}>0$ and $x>0$
$\Rightarrow \frac{\log _{10} x}{23-\log _{10} x} \geq 10^0=1$ and $\frac{\log _{10} x}{\log _{10 x^{-3}}}<0$ and $x>0$ $\Rightarrow \frac{\log _{10} x-23-\log _{10} x}{23-\log _{10} x} \geq 0$ and $\frac{\log _{10} x}{\log _{10} x-3}<0$
and
$x>0$
$\Rightarrow \frac{3 \log _{10} x-2}{2 \log _{10} x-3} \leq 0$ and $\frac{\log _{10} x}{\log _{10} x-3}<0$ and $x>0$
$\Rightarrow 2 \leq \log _{10} x<3$ and $0<\log _{10} x<3$ and $x>0$
$\Rightarrow 10^2 \leq x<10^3$ i.e. $100 \leq x<1000$
$\Rightarrow m=100, n=1000$
$\Rightarrow \frac{n}{m}=10$