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The least value of the expression \( \Large 2\ log_{10}x\ -\ logx\ \left(0.01\right) \) for x > 1, is:

A) 10
B) 2
C) -0.01
D) none of the above

Correct Answer:




D) none of the above

Description for Correct answer:

Here, \( \Large 2log_{10}x-log_{x} \left(10\right)^{-2}=2log_{10}x+2log_{x}10 \)

= \( \Large 2log_{10}x+2\frac{1}{log_{10}x} \)

= \( \Large 2\{ log_{10}x+\frac{1}{log_{10}x} \} \)

Using \( \Large AM \ge GM \) we get

\( \Large \frac{log_{10}x+\frac{1}{log_{10}x}}{2}\ge \left(log_{10}x\frac{1}{log_{10}x}\right)^{\frac{1}{2}} \)

=> \( \Large log_{10}x+\frac{1}{log_{10}x}\ge 2 \)

\( \Large \therefore 2log_{10}x-log_{x} \left(0.01\right)\ge 4 \)

Therefore, Least value is 4.

Part of solved Logarithms questions and answers : >> Elementary Mathematics >> Logarithms

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