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If the ${5.3}^{2 \log_{2}^{x}} - {30.7}^{\log_{7}^{x}} + 2^{\log_{2}^{6}}$ has minimum value at $x = a$ then $a$ is

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Detailed Solution

${5.3}^{2 \log_{2}^{x}} - {30.7}^{\log_{7}^{x}} + 2^{\log_{2}^{6}}$

$\Rightarrow {5.3}^{\log_{3} x^{2} - 30 x + 6}$

$\Rightarrow 5. x^{2} - 30 x + 6$

$a = 5 (> 0)$ minimum value at $x = a$

$x = \frac{- b}{2 a}$

$a = \frac{- (- 30)}{2 (5)} = \frac{30}{10} = 3$

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