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QJEEAdv22Mae166: JEE Advanced 2022 Mathematics - Class 11 JEEAdv22Mae166 Solution: The product of all positive real values of satisfying the…

The product of all positive real values of xx satisfying the equation

x(16(log⁡5x)3−68log⁡5x)=5−16x^{\left(16\left(\log _{5} x\right)^{3}-68 \log _{5} x\right)}=5^{-16}

is __________.

Step-by-Step Verified Solution

Taking log to the base 5 on both sides

(16(log⁡5x)3−68(log⁡5x))(log⁡5x)=−16\left(16\left(\log _{5} x\right)^{3}-68\left(\log _{5} x\right)\right)\left(\log _{5} x\right)=-16

Let (log⁡5x)=t\left(\log _{5} x\right)=t

16t4−68t2+16=016 t^{4}-68 t^{2}+16=0

⇒\Rightarrow 4t4−16t2−t2+4=04 t^{4}-16 t^{2}-t^{2}+4=0

⇒\Rightarrow (4t2−1)(t2−4)=0\left(4 t^{2}-1\right)\left(t^{2}-4\right)=0

⇒\Rightarrow t=±12,±2t=\pm \frac{1}{2}, \pm 2

So log⁡5x=±12\log _{5} x=\pm \frac{1}{2} or ±2\pm 2

⇒x=512,5−12,52,5−2\Rightarrow x=5^{\frac{1}{2}}, 5^{\frac{-1}{2}}, 5^{2}, 5^{-2}

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