Question

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 ______

Answer 1

$x^{16\left(\log _5 x\right)^3-68 \log _5 x}=5^{-16}$
Take $\log$ to the base 5 on both sides and put $\log _5 x=t$
$16 t ^4-68 t ^2+16=0 $
$ \Rightarrow 4 t ^4-17 t ^2+4=0 \begin{cases} t _1 \\t _2 \\t _3 \\t _4\end{cases} $
$ 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$