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

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 ${\log }_{5}x=t$
$16t^4-68t^2+16=0$
$\Rightarrow 4t^4-17t^2+4=0\{\begin{array}{l}{t}_1\\ t_2\\ t_3\\ t_4\end{array}$
$t_1+t_2+t_3+t_4=0$
${\log }_5{x}_1+{\log }_5{x}_2+{\log }_5{x}_3+{\log }_5{x}_4=0$
$x_1{x}_2{x}_3{x}_4=1$