Find the least value of the expression $2 l o g_{10} x - l o g_{x} 0.01$ ; where $x > 0, x \neq 1.$

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Solution

Let, $y = 2 l o g_{10} x - l o g_{x} 0 \cdot 01 = 2 (l o g_{10} x + l o g_{x} 10)$
Since, A.M $\geq$ GM
Therefore, $\frac{l o g_{10} x + l o g_{x} 10}{2} \geq \sqrt{l o g_{x} 10 l o g_{10} x} = 1$
$\Rightarrow y \geq 4$
Ans: 4

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