The product of all positive real values of x | vedclass

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Question

$x^{16\left(\log _5 x\right)^3-68 \log _5 x}=5^{-16}$

Take log to the base 5 on both sides and put $\log _5 x=t$

$16 t^4-68 t^2+16=0$

$\Right

Vedclass · Accepted Answer

$1$

Vedclass · Answer

$x^{16\left(\log _5 xight)^3-68 \log _5 x}=5^{-16}$

Take log to the base 5 on both sides and put $\log _5 x=t$

$16 t^4-68 t^2+16=0$

$\Rightarrow 4 t^4-17 t^2+4=0\left\{\begin{array}{l}t_1 \ t_2 \ t_3 \ t_4\end{array}ight.$

$t_1+t_2+t_3+t_4=0$

$\log _5 x_1+\log _5 x_2+\log _5 x_3+\log _5 x_4=0$

$x_1 x_2 x_3 x_4=1$

The product of all positive real values of $x$ satisfying the equation $x^{\left(16\left(\log _5 x\right)^3-68 \log _5 x\right)}=5^{-16}$is. . . . .

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