1) Cos²x=1/2
а) Cosx = √2/2 б)Cosx = -√2/2
x = +-arcCos√2/2 + 2πk , k ∈Z x = +-arcCos(-√2/2) + 2πn , n ∈Z
x = +-π/4 + 2πk , k∈Z x = +-3π/4 + 2πn , n ∈Z
2) log₂²10 = log₂10 * log₂10 =log₂(2*5) * log₂(2*5) =
=(log₂2 + log₂5)(log₂2 + log₂5)=(1 + log₂5)² = 1 + 2log₂5 + log₂²5
log₂10*log₂5 = log₂(2*5) * log₂5=( log₂2 + log₂5)*log₂5 =
=(1 + log₂5)*log₂5 = log₂5 + log₂²5
числитель = 1 + 2log₂5 + log₂²5+ log₂5 + log₂²5 -2log₂²5= 1 +3log₂5
знаменатель = log₂10 + 2log₂5 = log₂(2*5)+ 2log₂5 =
= log₂2 + log₂5 + 2log₂5 = 1 + 3log₂5
ответ: 1
3) 7/(9ˣ -2) - 2/(3ˣ -1) ≥ 0
(7*3ˣ -7 - 2*9ˣ +4)/(9ˣ-2)(3ˣ -1) ≥ 0
(7*3ˣ -3 - 2*9ˣ )/(9ˣ-2)(3ˣ -1) ≥ 0
метод интервалов.
7*3ˣ -3 - 2*9ˣ = 0, ⇒ 3ˣ = t, ⇒7t -3 - 2*t² = 0, ⇒ t₁ = 1/2, t₂ = 3
-∞ 1/2 3 +∞
- + - знаки 7t -3 - 2*t²
3ˣ < 0 √3 < 3ˣ <3
∅ 1/2 < x < 1
9ˣ-2 = 0, ⇒ 9ˣ = 2, ⇒ x = log₉2
3ˣ -1= 0, ⇒ 3ˣ = 1, ⇒ x = 0
-∞ 0 1/2 1 log₉2 +∞
+ знаки √3 < 3ˣ <3
1) Cos²x=1/2
а) Cosx = √2/2 б)Cosx = -√2/2
x = +-arcCos√2/2 + 2πk , k ∈Z x = +-arcCos(-√2/2) + 2πn , n ∈Z
x = +-π/4 + 2πk , k∈Z x = +-3π/4 + 2πn , n ∈Z
2) log₂²10 = log₂10 * log₂10 =log₂(2*5) * log₂(2*5) =
=(log₂2 + log₂5)(log₂2 + log₂5)=(1 + log₂5)² = 1 + 2log₂5 + log₂²5
log₂10*log₂5 = log₂(2*5) * log₂5=( log₂2 + log₂5)*log₂5 =
=(1 + log₂5)*log₂5 = log₂5 + log₂²5
числитель = 1 + 2log₂5 + log₂²5+ log₂5 + log₂²5 -2log₂²5= 1 +3log₂5
знаменатель = log₂10 + 2log₂5 = log₂(2*5)+ 2log₂5 =
= log₂2 + log₂5 + 2log₂5 = 1 + 3log₂5
ответ: 1
3) 7/(9ˣ -2) - 2/(3ˣ -1) ≥ 0
(7*3ˣ -7 - 2*9ˣ +4)/(9ˣ-2)(3ˣ -1) ≥ 0
(7*3ˣ -3 - 2*9ˣ )/(9ˣ-2)(3ˣ -1) ≥ 0
метод интервалов.
7*3ˣ -3 - 2*9ˣ = 0, ⇒ 3ˣ = t, ⇒7t -3 - 2*t² = 0, ⇒ t₁ = 1/2, t₂ = 3
-∞ 1/2 3 +∞
- + - знаки 7t -3 - 2*t²
3ˣ < 0 √3 < 3ˣ <3
∅ 1/2 < x < 1
9ˣ-2 = 0, ⇒ 9ˣ = 2, ⇒ x = log₉2
3ˣ -1= 0, ⇒ 3ˣ = 1, ⇒ x = 0
-∞ 0 1/2 1 log₉2 +∞
+ знаки √3 < 3ˣ <3