Question

Уравнение log 2 (x) - 2log x (2) = -1

Answer 1

Log(2)x -2log(x)2=-1
ОДЗ x>0 U x≠1⇒x∈(0;1) U (1;∞)
log(2)x-2/log(2)x =-1
loj(2)²x+log(2)x-2=0
log(2)x=a
a²+a-2=0
a1+a2=-1 U a1*a2=-2
a1=-2⇒log(2)x=-2⇒x=1/4
a2=1⇒log(2)x=1⇒x=2

Answer 2

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

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