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The least value of the expression 2 _{ 10 }{ x } - _{ 10 }{ (

Question

The least value of the expression $2 lo g_{10} x - lo g_{10} (0.01)$ for $x > 1$ ,

10

2

- 0.01

none of these

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The least value of the expression

2 lo g_{10} x - lo g_{10} (0.01)

for

x > 1

5 mins ago

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5 mins ago

Text solutionVerified

2 lo g_{10} x - lo g_{x} 0.01 = 2 lo g_{10} x - \frac{lo g _{10} 0.01}{lo g _{10} x} = 2 lo g_{10} x + \frac{2}{lo g _{10} x}

= 2 [lo g_{10} x + \frac{1}{lo g _{10} x}] [h ere x > 1 \Rightarrow lo g_{10} x > 0]

Now

A . M \geq G . M

\Rightarrow \frac{lo g _{10} x + \frac{1}{lo g _{10} x}}{2} \geq (lo g_{10} x \frac{1}{lo g _{10} x})^{1/2}

\Rightarrow lo g_{10} x + \frac{1}{lo g _{10} x} \geq 2

The least value is

2

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Question Text	The least value of the expression $2 lo g_{10} x - lo g_{10} (0.01)$ for $x > 1$ ,
Answer Type	Text solution:1
Upvotes	150