Question

Find the minimum value of the expression ?

$2{\log }_{10}x-\log 0.01;$ where $x>1$

Answer 1

Given,

$2{\log }_{10}x-\log 0.01,whenx>1=2{\log }_{10}x+2\log 10$

Here, as $x>1,{\log }_{10}x>0$

So, using $AM-GM$ inequality,

$\therefore AM\ge GM\Rightarrow \frac{2{\log }_{10}x+2\log 10}{2}\ge \sqrt{2{\log }_{10}x+2\log 10}\Rightarrow 2{\log }_{10}x-2\log 10\ge 4$

Hence, minimum value of $2{\log }_{10}x-\log 0.01$ is $4$.