Question

The greatest value of $(4{\log }_{10}x-{\log }_x(.0001\left)\right)for0⟨x⟨1$

• A. 4
• B. -4
• C. 8
• D. -8

Answer 1

Answer: (C)

8

$y=4{\log }_{10}x-{\log }_{x}0.0001$ for $0<x<1$

$y=4{\log }_{10}x+{\log }_x{10}^{-4}$

$y=4{\log }_{10}x+4{\log }_{x}10$

$y=4({\log }_{10}x+{\log }_{x}10)$

Differentiating and putting it equal to zero ,

$y\prime =4(\frac{1}{x}-\frac{1}{(\log x{)}^2}\frac{1}{x})=0$

$\Rightarrow y\prime =4\frac{\left[\right(\log x{)}^2-1]}{x(\log x{)}^2}=0$

$\Rightarrow y\prime =(\log x{)}^2=1$

$\therefore \log x=1$

$\Rightarrow x=10$

And, $\log x=-1$

$\Rightarrow x={10}^{-1}$

$y\left(10\right)=4({\log }_{10}10+{\log }_{10}10)=4(1+1)=8$

and $y\left({10}^{-1}\right)=4[{\log }_{10}{10}^{-1}+{\log }_{1/10}10]=-8$

Greater value of y is $8$