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 $4$

$2{log}_{10}x-{log}_x\left(0.01\right)=2{log}_{10}x+2{log}_{x}10\Rightarrow \frac{{log}_{10}x+{log}_{x}10}{2}\ge \sqrt{\left({log}_{10}x\right)\left({log}_{x}10\right)}(AM\ge GM)\Rightarrow {log}_{10}x+{log}_{x}10\ge 2\Rightarrow 2{log}_{10}x-{log}_{x}0.01\ge 4$

Least value is $4.$