task/28237162
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1.
{ Log₂x +Log₂y =4 ,
{y -x =6 .
ОДЗ: { x > 0 , y >0.
{ xy =2⁴ , { x(x+6) =16 , {x²+6x -16 =0 , {(x+8)(x-2) =0
{ y= x+6 . { y =x+6. { y =x+6 . { x =y+6
x² +6x -16 =0 * * * x² - ( (-8) + 2)x +(-8)*2 =0 ⇒ x₁ =-8, x₂=2. * * *
D₁ =3² -(-16) =25 =5²
x₁ =(-8-5 = - 8 →посторонний корень,
x₂=-8 +5 =2. ⇒ y₂= x₂+6 =2+6 =8.
ответ: x =2 , y =8. * * * ( 2 ; 8) * * *
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2.
ОДЗ: { x> 0 , y>0.
{ Log₂x +Log₂y = 2+ Log₂5 ; { Log₂x +Log₂y = Log₂4+ Log₂5 ,
{Log₂(x-y) =0 . { {Log₂(x-y) =Log₂1 .
{Log₂ xy = Log₂4*5 , {xy =20 , {x(x- 1) -20 =0 , {x² - x - 20 =0 ,
{x-y =1. { y =x-1. { y =x-1. { y =x-1.
x² - x - 20 =0 ⇒x₁ =(1-9)/2 = - 4 → посторонний корень,
x₂=(1+9)/2 =5. ⇒y₂= x₂- 1 =5 -1 =4.
ответ : ( 5 ; 4) .
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3.
Log₄(x -1) + Log(√2) (x -1) < 2, 5 ;
ОДЗ: x -1 >0 ⇔ x ∈(1; ∞)
(1/2)*Log₂(x -1) + 2*Log₂ (x -1) < 2, 5 ;<br>2,5*Log₂ (x -1) < 2, 5 ;<br>Log₂ (x -1) < 1;<br>Log₂ (x -1) < Log₂ 2 ⇔ 0 < x -1< 2 ⇔ 1
ответ : x ∈( 1 ; 3)