Question

The least value of the expression $2{\log }_{10}x-{\log }_x\left(0.01\right)$ for $x>1$ is:

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is D $4$

Given,

$2{\log }_{10}x-{\log }_x{10}^{-2}$

$=2{\log }_{10}x+2{\log }_{x}10$

$=2{\log }_{10}x+2\left(\frac{1}{{\log }_{10}x}\right)$

using $AM\ge GM$ we get,

$\frac{{\log }_{10}x+\left(\frac{1}{{\log }_{10}x}\right)}{2}\ge {\left({\log }_{10}x\frac{1}{{\log }_{10}x}\right)}^{\frac{1}{2}}$

${\log }_{10}x+\left(\frac{1}{{\log }_{10}x}\right)\ge 2$

or

$2{\log }_{10}x-{\log }_x\left(0.01\right)\ge 4$