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

Question

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 ________.

Smart Achievers · Accepted Answer

(1)

$We have,$ $x^{(16 {(\log_{5} x)}^{3} - 68 \log_{5} x)} = 5^{- 16}$

Taking log both sides and take $\log_{5} x = t$ , we get

$t (16 t^{3} - 68 t) = - 16$

$\Rightarrow$ $16 t^{4} - 68 t^{2} + 16 = 0 \Rightarrow 4 t^{4} - 17 t^{2} + 4 = 0$

$∴$ $t_{1} + t_{2} + t_{3} + t_{4} = 0$

$\Rightarrow$ $\log_{5} x_{1} + \log_{5} x_{2} + \log_{5} x_{3} + \log_{5} x_{4} = 0 \Rightarrow x_{1} x_{2} x_{3} x_{4} = 1$

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