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 ________. [2022]

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 ________. [2022]

Smart Achievers · Accepted Answer

(1)

Taking log with base 5 on both sides

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

$Let \log_{5} x = t$

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

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

$\Rightarrow (4 t^{2} - 1) (t^{2} - 4) = 0$

$\Rightarrow t = \pm \frac{1}{2}, \pm 2$

$So \log_{5} x = \pm \frac{1}{2}, \pm 2$ $\Rightarrow x = 5^{1 / 2}, 5^{- 1 / 2}, 5^{2}, 5^{- 2}$

$Product = 5^{1 / 2} \cdot 5^{- 1 / 2} \cdot 5^{2} \cdot 5^{- 2} = 1$

Solution

Q.
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 ________. [2022]

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Solution

Q. The product of all positive real values of x satisfying the equation x(16(log5x)3-68log5x)=5-16 is ________. [2022]

Ans. (1) Taking log with base 5 on both sides (16(log5x)3-68(log5x))(log5x)=-16 Let log5x=t ⇒16t4-68t2+16=0 ⇒4t4-17t2+4=0 ⇒(4t2-1)(t2-4)=0 ⇒t=±12, ±2 So log5x=±12, ±2⇒x=51/2, 5-1/2, 52, 5-2 Product =51/2·5-1/2·52·5-2=1

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Q.
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 ________. [2022]