Question

The least value of the expression $2{\log }_{10}x-{\log }_{x}0.01$ is...........

Answer 1

The expression can be rewritten as

$2{\log }_{10}x-\frac{{\log }_{10}\left({10}^{-2}\right)}{{\log }_{10}x}$

$=2{\log }_{10}x+2\frac{{\log }_{10}\left(10\right)}{{\log }_{10}x}$

As we know $A.M\ge G.M$

$2{\log }_{10}x+2\frac{{\log }_{10}\left(10\right)}{{\log }_{10}x}\ge 2\sqrt{2{\log }_{10}x\ast 2\frac{{\log }_{10}\left(10\right)}{{\log }_{10}x}}$
$\Rightarrow 2{\log }_{10}x+2\frac{{\log }_{10}\left(10\right)}{{\log }_{10}x}\ge 4$

Thus, the least value of given expression is $4$.