Question

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

• A. None of these
• B. $10$
• C. $-0.01$
• D. $2$

Answer 1

The correct option is A None of these

Here, $2{\log }_{10}x-{\log }_x(10{)}^{(-2)}=2{\log }_{10}x+2{\log }_{x}10$
$$=2{\log }_{10}x+2\frac{1}{{\log }_{10}x}=2({\log }_{10}x+\frac{1}{{\log }_{10}x})...\left(i\right)$$
Using AM $\ge$ GM, we get
$$\begin{array}{cc}& \frac{{\log }_{10}x+\frac{1}{{\log }_{10}x}}{2}\ge {\left({\log }_{10}x\frac{1}{{\log }_{10}x}\right)}^{1/2}\\ & \Rightarrow {\log }_{10}x+\frac{1}{{\log }_{10}x}\ge 2...\left(ii\right)\end{array}$$
or $2{\log }_{10}x-{\log }_x\left(0.01\right)\ge 4$
Hence, least value is 4 .