Объяснение:
1. log₁₅(51x-1)-log₁₅x=0
ОДЗ: 51x-1>0 51x>1 |÷51 x>1/51 x>0 ⇒ x∈(1/51;+∞)
log₁₅(51x-1)=log₁₅x
51x-1=x
50x=1 |÷50
x=1/50.
2. log²₄x=log₄x⁷-12 ОДЗ: x>0.
log²₄x-7*log₄x+12=0
Пусть log₄x=t ⇒
t²-7t+12=0 D=1
t₁=log₄x=3 x=4³ x₁=64.
t₂=log₄x=4 x=4⁴ x₂=256.
3. log₀,₁v (v)+log₀,₂v (v)=0 ОДЗ: v>0 v≠10 v≠5 ⇒
v∈(0;5)U(5;10)U(10;+∞)
(1/log(v)0,1v)+(1/log(v)0,2v)=0
(log(v)0,2v+log(v)0,1v)/(log(v)0,2v*log(v)0,1v)=0
(log(v)0,2v*log(v)0,1v)≠0 ⇒
(log(v)0,2v+log(v)0,1v)=0
log(v)(0,2v*0,1v)=0
log(v)0,02v²=0
0,02v²=v⁰
0,02v²=1 |÷0,02
v²=50
v₁=√50 ∈ ОДЗ v₂=-√50 ∉ ОДЗ.
4. lglglog₅x=0
lglog₅x=10⁰
lglog₅x=1
log₅x=10¹
log₅x=10
x=5¹⁰ ⇒
¹⁰√5¹⁰=5.
5. 2*lgx/lg(5x-4)=1 ОДЗ: x>0 5x-4>0 5x>4 x>0,8 ⇒ x∈(0,8;+∞).
lgx²=lg(5x-4)
x²=5x-4
x²-5x+4=0 D=9 √D=3
x₁=1 x₂=4.
∑x₁,₂=1+4=5.
Объяснение:
1. y=log₇(x²+5x-6)
ОДЗ: x²+5x-6>0 x²+5x-6=0 D=49 √D=7 x₁=-6 x₂=1 ⇒
(x+6)(x-1)>0 -∞__+__-6__-__1__+__+∞ D(f)=(-∞;-6)U(1;+∞).
2. y=log₂(x+c) (2;3)
x+c=2^y
2+c=2^3
c=8-2
c=6.
3. y=log₁₁(11^(t-2)-121)
ОДЗ: 11^(t-2)-121>0 11^(t-2)>121 11^(t-2)>11² t-2>2 t>4 t∈(4;+∞)
4. 2^s=5 log₂(2^s)=log₂5 s*log₂2=log₂5 s=log₂5.
5. log₀,₄(15+2x)=1
ОДЗ: 15+2x>0 2x>-15 |÷2 x>-7,5 ⇒ x∈(-7,5;+∞).
15+2x=0,4¹ 2x=-14,6 |÷2 x=-7,3 ∈ ОДЗ
π/6+πn≤π/4-x<π/2+πn
π/6-π/4+πn≤-x<π/2-π/4+πn
-π/24+πn≤-x<π/4+πn
-π/4+πn<x≤π/24+πn
x∈(-π/4+πn;π/24+πn]
2)2πn≤x≤π+2πn
3π/4+2πn≤x≤5π/4+2πn
x∈[3π/4+2πn;π+2πn}
3)cosxcosy=1/4⇒cos(x-y)+cos(x+y)=1/2
ctgxctgy--3/4⇒1/4:sinxsiny=-3/4⇒sinxsiny=-1/3⇒cos(x-y)-cos(x+y)=-2/3
прибавим и отнимем
2сos(x-y)=-1/6⇒cos(x-y)=-1/12⇒x-y=π-argcos1/12
2cos(x+y)=7/6⇒cos(x+y)=7/12⇒x+y=arccos7/12
прибавим и отнимем
2x=π-arccos1/12+arccos7/12⇒x=π/2-1/2arccos1/12+1/2arccos7/16
2y=π-arccos1/12-arccos7/12⇒x=π/2-1/2arccos1/12-1/2arccos7/16
4)2sin2xsin4x=0
sin2x=0⇒2x=πn⇒x=πn/2
sin4x=0⇒4x=πn⇒x=πn/4
ответ x=πn/4