If the product of all positive real values of ‘x’ satisfying the equation $x^{(16 {(\log_{5} x)}^{3} - 68 \log_{5} x)} = 5^{- 16}$ is k; then which of the following is CORRECT about ‘k’

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$\log_{5} k = 2$

k satisfies $k^{2} - 3 k + 2 = 0$

‘k’ is composite

‘k’ is prime

answer is C.

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Detailed Solution

Let $\log_{5}^{x} = t$
Apply $\log_{5}$ on both sides
$(16 t^{3} - 68 t) t = - 16$
$\Rightarrow 16 t^{4} - 68 t^{2} + 16 = 0 \Rightarrow 4 t^{4} - 17 t^{2} + 4 = 0$
$\Rightarrow x = 5^{\pm 2}; 5^{\pm 1 / 2} =$ product = 1 = k

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