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

Answer: 10

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