If $a > 1, b > 1$ , then the minimum value of ${log}_{b} a + {log}_{a} b$ is

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none of these

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Solution

The correct option is C 2
As $a > 1, b > 1$ , $l o g_{a} b, l o g_{b} a$ are positive quantities.
$A M \geq G M$
$\Rightarrow l o g_{a} b + l o g_{b} a \geq 2 \sqrt{l o g_{a} b l o g_{b} a}$
$\Rightarrow l o g_{a} b + l o g_{b} a \geq 2 \sqrt{\frac{l o g_{e} b}{l o g_{e} a} \frac{l o g_{e} a}{l o g_{e} b}} = 2$
Hence, option C is correct

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