Уравнение log 2 (x) - 2log x (2) = -1

Решите задачу:

$log _{2} x-2log _{x} 2=-1 \\ log _{2} x- \frac{2}{log _{2} x} =-1 \\ x\ \textgreater \ 0 \\ x \neq 1\\ log _{2} x=t \\ t- \frac{2}{t} =-1 \\ \\ t- \frac{2}{t} +1=0 \\ \\ \frac{t ^{2}-2+t }{t} =0 \\ \\ t \neq 0 \\ \\$

$t ^{2} +t-2=0 \\ D=1+8=9 \\ \sqrt{D} =3 \\ t _{1} = \frac{-1+3}{2} = \frac{2}{2} =1 \\ t _{2} = \frac{-1-3}{2} = \frac{-4}{2} =-2 \\ \\ log _{2} x=1 \\ log _{2} x=log _{2} 2 \\ x=2 \\ \\ log _{2} x=-2 \\ x= \frac{1}{4}$