Question

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

• A. $10$
• B. $2$
• C. $-0.01$
• D. $4$

Answer 1

The correct option is D $4$

Let $L=2{\log }_{10}x-{\log }_x\left(0.01\right)$, $x>1$
$\Rightarrow L=2{\log }_{10}x-{\log }_x\left({10}^{-2}\right)$
$\Rightarrow L=2{\log }_{10}x+2{\log }_{x}10$
$\Rightarrow L=2({\log }_{10}x+\frac{1}{{\log }_{10}x})\dots \left[1\right]$
For $x>1$
${\log }_{10}x>0$ and $\frac{1}{{\log }_{10}x}>0$
Using A.M.-G.M. inequality, we have
$\frac{{\log }_{10}x+\frac{1}{{\log }_{10}x}}{2}\ge {({\log }_{10}x\times \frac{1}{{\log }_{10}x})}^{1/2}$
${\log }_{10}x+\frac{1}{{\log }_{10}x}\ge 2\dots \left[2\right]$
From $\left[1\right]$ and $\left[2\right]$, we get
$L\ge 4$
Therefore, the least value of the given expression is $4$.