If $x > 1$ , find the least value of the expression $2 {log}_{5} x - {log}_{x} (0.04)$

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Solution

Let, $y = 2 l o g_{5} x - l o g_{x} 0 \cdot 04 = 2 (l o g_{5} x + l o g_{x} 5)$
Since, A.M $\geq$ GM
Therefore, $\frac{l o g_{5} x + l o g_{x} 5}{2} \geq \sqrt{l o g_{x} 5 l o g_{5} x} = 1$
$\Rightarrow y \geq 4$
Ans: 4

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