Question

The minimum value of the expression$2{\log }_{10}x-{\log }_{x}0.01$ where $x>1,$ is ____________.

• A. $2$
• B. $0.1$
• C. $4$
• D. $1$

Answer 1

Answer: (C)

$4$

Given $f\left(x\right)=2{\log }_{10}x-lo{g}_{x}0.01$

$f\left(x\right)=2{\log }_{10}x-\frac{lo{g}_{10}{10}^{-2}}{{\log }_{10}x}=2\log x+\frac{2}{logx}$

For minima $f^{\prime }\left(x\right)=0$

$\Rightarrow \frac{2}{x}-\frac{2}{x\log x}=0$

$\Rightarrow \left(\frac{2}{x}\right)(1-\frac{1}{\log x})=0$

$\Rightarrow x=10$

Now,

$f\left(10\right)=2{\log }_{10}10-lo{g}_{10}0.01=2-(-2)=4$