Question

If $x>1$, then the least value of the expression $2{\log }_{10}x-{\log }_{x}0.01$ is

• A. $0$
• B. $1$
• C. $2$
• D. $4$

Answer 1

The correct option is D $4$

We have,
$2{\log }_{10}x-{\log }_{x}0.01$
$=2{\log }_{10}x-{\log }_x(10{)}^{-2}$
$=2{\log }_{10}x+2{\log }_{x}10$
$=2({\log }_{10}x+\frac{1}{{\log }_{10}x})=2[{(\sqrt{{\log }_{10}x}-\sqrt{\frac{1}{{\log }_{10}x}})}^2+2][\because x>1\Rightarrow {\log }_{10}x>0]$

Hence, the minimum value is $4$