Practice - Exponential equations Restriction k >0 Equ at io n Typ e 1
1. 2.
2
0,5 x ⋅ 2 2 x + 2 =
5. 6. 7. 8. 9. 10 .
x
5
1 = 5
0,125 ⋅ 4 3 4
x +1
−3 x − 4
2 x −3
2 = 8
−x
2
9 4x ⋅ = 16 3
D=R \ {0}
3 x − 4 ⋅ 27 3− 2 x = 9 3 x −3 9
0 , 25 x 2 − 2 x −8
3
−x 3
4
3 5
12 . 13 . 14 .
2x
2
6 ⋅ 5
6x −4
2− x
=
= 64
2 x − 4 ⋅ 8 3− 2 x = 4 3 x − 3
4
x +5
Z={-2,4}
-x+ =4
1 2
Z={- 3 }
-x2+3x+4=0
Z={-1,4}
3x=18
Z={6}
x2-x-2=0
Z={-1,2}
3 −5 x +5 = 36 x −6
Z={1} Z={-3,11}
1 2
Z={- 3 ,0, 3 } Z={2}
x2-5x+6=0
Z={2,3}
x2-5x+4=0
Z={1,4}
-11x=-11
Z={1}
x −1 2 = x +1 3
1−x
3
x2-2x-8=0
25 36
=1
1 x +1 = 2
4 3
Z={ }
x -8x-33=0 -x3+3x=0
x 2 −5 x + 6
− 5 x +10
Answe r
2
= 3
3 ⋅9 = 3
5x 6
11 .
15 .
1 64
1 3 ⋅ = 81 3 x2
Hints
3x=4
3 5 x −8 = 9 x − 2
3. 4.
Su bst itu ti on
D=R \ {-1}
Z={5} Z={10}
x +17
32 x − 7 = 0,25 ⋅ 128 x −3 Typ e 2
16 . 17 . 18 .
3 x + 2 − 3 x = 72
3x=k
2 x +3 − 2 x = 112 2 x − 2 x −4 = 15
8k=72 7 ⋅ 2 x = 112
2x=k
k=16
k=9 2x=16
Z={2} Z={4} Z={4}
1
19 . 20 . 21 . 22 . 23 . 24 . 25 . 26 . 27 . 28 . 29 . 30 . 31 . 32 . 33 .
5 x + 3 ⋅ 5 x − 2 = 140
5x=k
k=125
Z={3}
3 2 x + 2 + 3 2 x = 30
32x=k
k=3
Z={ }
3 2 x −1 + 3 2 x − 2 − 3 2 x − 4 = 315
32x=k
35k=315*81 k=36
Z={3}
2 x + 2 − 2 x −1 = 14
2x=k
Z={2}
3 x +1 − 3 x − 3 x −1 = 15
3x=k
Z={2}
4
x−2 2
−2
x +1
x +1 3
=8
3 2x +
4 = 2 ⋅ 3 2 x +1 − 9 x 27
Z={2}
1 x ⋅ 2 − 4 ⋅ 2 x + 15 = 0 4
− 15
8 ⋅ 5 x + 7 ⋅ 5 x −1 = 22 + 5 x +1
1 2
5x=k
Z={1}
32x=k
7 ⋅ 4 x − 2 2 x +1 = 26 + 7 ⋅ 4 x −1
4x=k
3 ⋅ 2 x − 20 = 2 x −1
2x=k
1 2
Z={ − 1 } 1 2
Z={ 1 } Z={3} Z={1}
7 x + 2 + 2 ⋅ 7 x −1 = 345 2 x +1 + 3 ⋅ 2 x −1 − 5 ⋅ 2 x + 6 = 0
2x=k
5 ⋅ 2 3 x + 8 x − 2 ⋅ 41,5 x = 16
23x=k
2x 2
2
2
− x+2
+ 5 ⋅ 2x
x −3 x
1 − 5⋅ 2
2+
− x −1
= 26
3− x +2 x
= 11
2x 2
2
2 3
Z={ }
=k
Z={-1,2}
=k
Z={-3}
−x
x −3 x
Z={2}
Typ e 3
34 . 35 . 36 . 37 . 38 . 39 . 40 .
k2+3k-108=0 k1=9 ; k2=-12
3 x +1 + 9 x = 108
3x = k
2 ⋅ 16 x − 17 ⋅ 4 x + 8 = 0
4x=k
4x − 9 ⋅ 2x + 8 = 0
2x=k
k2-9k+8=0 k1=8 ; k2=1
Z={0,3}
4x − 8⋅ 2x = 0
2x=k
k2-8k=0
Z={3}
2 x +1 + 4 x = 80
2x=k
k2+2k-80=0 k1=-10 ; k2=8
Z={3}
4x + 8 = 6 ⋅ 2x
2x=k
k2-6k+8=0 k1=2 ; k2=4
Z={1,2}
4 x − 10 ⋅ 2 x −1 = 24
2x=k
k2-5k-24=0 k1=-3 ; k2=8
Z={3}
2k2-17k+8=0 k1=8 ; k2=
Z={2} 1 2
3 2
1 2
Z={ ,− }
2
41 4x 5⋅ 2x − =8 . 2 42 8 ⋅ 3 x − 2 + 1 = 9 x −1 . 43 9x-2·3 x −1 -7=0 . 44 9 x − 486 = 3 x + 2 . 45 4 2 x +1 = 65 ⋅ 4 x −1 − 1 . 46 2 2 x + 2 − 3 ⋅ 2 x + 2 + 8 = 0 . 47 3 x + 2 + 9 x +1 = 810 . 48 25 x − 5 x +1 + 5 = 5 x . 49 3 2 x −1 + 3 x +1 = 12 . 50 5 2 x −1 + 5 x +1 = 250 . x x 51 2 2 3 ⋅ 4 − 7 ⋅ 2 = 20 . 2 52 1 3 x − x + 3⋅8 3 − 4 = 0 . 2
2x=k
k3+3k2-4=0; k1=1 ; k2=-2
53 . 54 . 55 .
30 − 5 x = 5 − x +3
1 =k 2
5x=k
-k2+30k-125=0 k2=5
9
3x=k
3x + 1 5
3
2x
x +1
=
3x=k 3k2-2k-21=0 ; k1=-
7 3
; k2=3
Z={3}
4x=k
Z={-2,1}
2x=k
Z={1,0}
3x=k
k2+k-90=0 k1=-10 ; k2=9
Z={2}
5x=k
k2-6k+5=0 k1=1 ; k2=5
Z={0,1}
3x=k
k1=-12 ; k2=3
Z={1}
5x=k
k2+25k-1250=0 k2=-50
x
k1=-
x
k1=25 ; Z={2} Z={4}
5 k2=4 3
3k2-28k+3=0 k1=
Z={0}
k1=25 ; Z={2,1} 1 ; k2=9 3
Z={-1,2} Z={-2}
x
=0
Z={1}
3x=k
22 = k
−2
Z={1,3} Z={2}
3x=k
28 3
1 − 24 ⋅ 5 − x − 5
k2-10k+16=0 k1=2 k2=8
1 =k 5
Typ e 4
56 11x −7 = 17 7 − x . 57 7 ⋅ 3 x +1 − 5 x + 2 = 3 x + 4 − 5 x +3 .
11x −7 =
1 17 x −7
(11 ⋅ 17 ) x −7
=1
Z={7}
3 =k 5
Z={-1}
2x=k
Z={0,2}
x
Bo nus
58 . 59 . 60 .
1 + 21− x = 1 x 2 −2
(2 + 3 ) + (2 − 3 ) x
4
3x
−7⋅4
2x
x
=4
+ 14 ⋅ 4 − 8 = 0 x
(2 + 3 ) x
4 =k
x
=k
k2-4k+1=0 k1= 2 − 3
k2= Z={1,-1}
2+ 3 1 2
Z={ 0, ,1 }
3
61 .
4 sin
62 . 63 .
2 2 sin x cos x = 2
2
1 2
x
= 2 ⋅ 2 cos x
sin2 x
=
1 2
dla cosx=-1
wynik: x1= ∏ +2k ∏ ;
1 2 1 sin2x= 2
∏ ∏ + 2k ∏ ; x3= − + 2k ∏ 3 3 ∏ 5 x1= + k ∏ ; x2= ∏ +k ∏ 12 12
dla cosx=
wynik: x2=
dla
wynik:
dla sinx= dla sinx=-
2 2 2 2
wynik: x1=
∏ 3∏ + 2k ∏ ;x2= + 2k ∏ ; 4 4
wynik: x3=-
∏ 5 + 2k ∏ ;x4= ∏ +2k ∏ 4 4
4