The least value of $2 {log}_{100} a - {log}_{a} 0.0001, a > 1$ is

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none of these

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Solution

The correct option is C 4
$2 l o g_{100} a - l o g_{a} 10^{- 4}$ $= 2 \frac{l o g_{10} a}{l o g_{10} 100} + 4 \frac{l o g_{10} 10}{l o g_{10} a}$
$= 2 \frac{l o g_{10} a}{2} + 4 \frac{1}{l o g_{10} a}$
For the two positive numbers $l o g_{10} a$ and $\frac{4}{l o g_{10} a}, A M \geq G M$
$\Rightarrow \frac{l o g_{10} a + \frac{4}{l o g_{10} a}}{2} \geq \sqrt{l o g_{10} a . \frac{4}{l o g_{10} a}} = 2$
$∴ l o g_{10} a + \frac{4}{l o g_{10} a} \geq 4$
Hence, option C is correct

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