Task/27552255
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1.
а) 5 +1 -2 =4 .
б) Log₂(√5 - 1)(√5 - 1)Log₃49 /Log₃7 =Log₂( (√5)² - 1²)Log₃7²/Log₃7 =
Log₂(5 - 1) * 2Log₃7/Log₃7 =Log₂2² * 2 =2*2=4 .
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2.
а) (1/9)ˣ +8*(1/3)ˣ - 9 =0 ⇔(1/3)²ˣ +8*(1/3)ˣ - 9 =0 ⇔( (1/3)ˣ +9)( (1/3)ˣ -1) =0 ⇔[ (1/3)ˣ +9=0 , (1/3)ˣ - 9 =0 ⇔ (1/3)ˣ =(1/3)⁻² ⇔ x = -2.
* * * (1/3)ˣ +9 > 9 ≠ 0 * * *
б) Log₃ x +4Log₉ x =9 ⇔Log₃ x +4Log₃² x =9⇔Log₃ x +4*(1/2)Log₃ x =9⇔
Log₃ x +2Log₃ x=9⇔3Log₃ x=9⇔Log₃ x=3 ⇔x =27.
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3.
а) 2ˣ(2³ -3*2¹+1) < 12 ⇔3*2ˣ < 12 ⇔2ˣ < 2² ⇔ x < 2. * * * 2 >1* * *
иначе x ∈ ( -∞ , 2) .
б) (Log₂⁻¹ x)² -3*Log₂⁻¹ x - 4 ≤ 0 ⇔(Log₂ x)² +3*Log₂ x - 4 ≤ 0 ⇔
(Log₂ x)² +3*Log₂ x - 4 ≤ 0 ⇔((Log₂ x +4)(Log₂ x - 1) ≤ 0 ⇔
-4 ≤ Log₂ x ≤ 1 ⇔2⁻⁴ ≤ x ≤ 2¹ ⇔ x ∈ [ - 4 , 2] .
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4.
Докажите числовое неравенство
(√3) ^ ( Log₃ (√5 -2)² ) +(√2) ^ ( Log₂(√5 -3)² ) =1.
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(√3) ^ ( Log₃ (√5 -2)² ) +(√2) ^ ( Log₂(√5 -3)² ) =
(√3) ^ (2*Log₂|√5 -2| ) +(√2) ^ ( 2Log₂|√5 -3| ) =
(√3)² ^ Log₃(√5 -2) +(√2)² ^ Log₂(3 -√5) =
(3) ^ Log₃(√5 -2) ) +(2) ^ ( Log₂(3 -√5 ) = (√5 -2) +(3 -√5 ) =1.