1
Logaritma 1.
2
log 4+ 2 log12− 2 log 6 = ........
Jawab : 2
2.
log
4.12 2 = log 8 = 3 6
Jika 2 log 7 = a maka tentukan 8 log 49 Jawab : 8
3.
3
log 49= 2 log 7 2 = 23 .2 log 7 =
2 3
a
Jika 5 log 3 = a dan 3log 4 = b maka tentukan 12 log 75 Jawab :
1 1 1 1 + 3 = 25 + 3 5 log12 log12 log 3+ log 4 log 3+ 3 log 4 2 1 2 1 2+ a = 5 + 3 = + = 5 3 log 3+ log 3. log 4 1+ log 4 a + ab 1 + b a(1 + b)
12
4.
log 75= 12 log 52 + 12 log 3 = 2 5
Tentukan nilai x yang memenuhi persamaan
(3 x + 2)
log 27 = 5 log 3
Jawab : (3 x + 2)
5.
3
log 27 = 5 log 33 ⇒ 3 x + 2 = 125 ⇔ x = 41
Jika x ≠ 1 dan x > 0, maka tentukan nilai x yang memenuhi persamaan x
log( x + 12) − 3.x log 4 + 1 = 0
Jawab : x
log( x + 12)+ x log x = x log 43 ⇔ x log( x 2 + 12 x )= x log 64
x 2 + 12 x − 64 = 0 ⇔ ( x + 16)( x − 4) = 0 x = − 16 tidak memenuhi x= 4
6.
Jika x1 dan x2 memenuhi persamaan (4 − log x) log x = log1000 maka tentukan x1 x2 Jawab :
a ( p log x) 2 + b( p log x) + c = 0 ⇒ x1 x2 = p
−
b a
−4
− log 2 x + 4 log x − 3 = 0 ⇒ x1 x2 = 10 − 1 = 10.000 7.
Tentukan penyelesaian pertaksamaan 6 log( x 2 − x) < 1 Jawab : 6
log( x 2 − x)< 6 log 6 ⇒ ( x − 3)( x + 2) < 0 ⇔ − 2 < x < 3 .........(1)
Syarat : x 2 − x > 0 ⇔ x( x − 1) > 0 ⇔ x < 0 atau x > 1 ............( 2) (1) ∩ (2) ⇒ − 2 < x < 0 atau 1 < x < 3
2
8.
a
1 1 1 log .b log 2 .c log 3 = ....... b c a
Jawab :
a
9.
a
log b − 1.b log c − 2 .c log a − 3 = − 1.(− 2).(− 3).a log b.b log c.c log a = − 6
log 3 a .a log a a = .......
Jawab : a
1
[ ]
maka tentukan ( a 2 ) 3 −
10. Jika 2 log 3 = a Jawab : 2
1 2
1 2
log 3 = a ⇔ a = 23
[( a ) ]
− 2 3
11. Jika
3
log a 3 .a log a 2 = 13 . 32 .a log a.a log a =
[
1 2
= ( 23 ) 6
]
−
1 2
=
1 512
a = 0,16666…….. maka tentukan a log 36
Jawab :
100a = 16,666….. 10a = 1,666…..
-
1 90a = 15 ⇔ a = 6 a
1
log 36= 6 log 36 = − 2
12. Jika a = 0,111…..
dan b = 0,333….. maka tentukan a log b
Jawab :
a = 0,111...... = b = 0,333..... = a
13. Jika log
1
log b = 9 log 13 =
1 9 1 3 1 2
a2 b = 12 maka tentukan log 3 2 b a
Jawab :
log
a2 a a = 12 ⇔ 2 log = 12 ⇔ log = 6 2 b b b
log 3
b 1 b 1 a = log = log a 3 a 3 b
−1
= −
1 a 1 log = − .6 = − 2 3 b 3
2
14. Jika
3 log a log a m = m dan = n, a > 1 dan b > 1 maka = ....... 3 2 log b log b n
Jawab :
m = n
2
log a 2 log b . = 3 log b 3 log a
a
log 3 b log 3 2 . = log 3.2 log 3 = ( 2 log 3) 2 a log 2 b log 2
3
15. Jika a = Jawab :
(
)(
)(
)
1 2 9 a maka 2 log 6 3 log 5 5 log 2 = ......... 5 3 6. 3
2
(
5 5 log 2 = 6. 3 2 .3 log 5 5 5 log 2 1
log 5
1
)
−1
(
= 6. 3
3
log 5
) (5 1 2
5
log 2 − 1
)
1
= 6.(5 2 ).(2 − 1 ) = 3 5
16. 5 log 27 .9 log125+ 16 log 32 = .......... Jawab : 5
17. Jika
a
3
2
4
log 3 2 .3 log 53 + 2 log 25 =
. . log 3.3 log 5 + 54 .2 log 2 =
3 3 5 2 2
9 4
+
5 4
=
7 2
log 3= b log 27, a > 0, b > 0, a ≠ 1, b ≠ 1 maka a log b = ........
Jawab :
log 3 log 27 log b log 27 a = ⇔ = ⇔ log b = 3 log 27 = 3 log a log b log a log 3
18. 2.3 log 4 − 12 .3 log 25+ 3 log10− 3 log 32 = .......... Jawab : 3
19.
log
4 2.10 1 2
25 .32
= 3 log1 = 0
log(5 5 ) + log 3 + log 45 = ......... log15 Jawab : 5 log 5 5. 3.45 15 5 = log15 2 = log15 2
( 20.
3
) ( 2
log 36 − 3 log 4 3 log 12
Jawab :
21.
(
(
3
)
2
= .........
)(
)
log 36+ 3 log 4 3 log 36− 3 log 4 = 1 3 . log12 2
)
( )
log x x + log y + log xy 2 = ........ log ( xy ) Jawab :
(
)
log x x . y .xy 2 log ( xy ) 2 5 = = log ( xy ) log ( xy ) 2 5
3
log144.3 log 9 2.23 log12 = 13 = 8 1 3 . log12 . log12 2 2
4
22. Jika a log 81 − 2.a log 27 + a log 27 + a log 243 = 6
maka tentukan a !
Jawab : a
log
81.27.243 = 6 ⇔ a 6 = 729 = 36 ⇒ a = 3 27 2
23. Jika 9 log 8 = 3m
maka tentukan 4 log 3
Jawab : 32
4
24. Jika
2
log 23 = 3m ⇔
. log 2 = 3m⇔ 3 log 2 = 2m
3 3 2
2
log 3= 2 log 3 = 12 .2 log 3 =
1 2
.3
1 1 1 1 = . = log 2 2 2m 4m
log 3 = a dan 3 log 5 = b maka 4 log 45 = .........
Jawab : 4
2
2
log 45= 2 log 32 + 2 log 5 =
22 1 1 1 . log 3 + .2 log 5= 2 log 3 + ( 2 log 3.3 log 5) = a + (ab) 2 2 2 2
25. Jika 7 log 2 = a dan 2 log 3 = b maka 6 log 98 = ...... Jawab : 6
2 1 + 2 7 log 2+ log 3 log 3+ 2 log 2 2 1 2 1 a+ 2 = 7 + 2 = + = 7 2 log 2+ log 2. log 3 log 3 + 1 a + ab b + 1 a (b + 1)
log 98= 6 log 7 2 + 6 log 2 = 2. 7
1 + log 6
2
1 = log 6
7
( x + 2 y )2 = ........ 26. Jika x > y > 1 dan x + 4 y = 12 xy maka log ( x − 2 y )2 2
2
Jawab :
i. x 2 + 4 y 2 + 4 xy = 16 xy ⇔ ( x + 2 y ) 2 = 16 xy ii. x 2 + 4 y 2 − 4 xy = 8 xy ⇔ ( x − 2 y ) 2 = 8 xy log
( x + 2 y)2 16 xy = log = log 2 2 ( x − 2 y) 8 xy
27. Diketahui 2 log 3 = 1,6 dan 2 log 5 = 2,3 . Tentukan nilai 2 log Jawab : 2
28. Jika
8
log
125 9
125 2 = log 53 − 2 log 32 = 32 log 5 − 22 log 3 = 3.2,3 − 2.1,6 = 3,7 9
log 5 = r maka 5 log16 = ........
Jawab : 8
log 5 = r ⇔
. log 5 = r ⇔ 2 log 5 = 3r
5
log16 = 4.5 log 2 =
1 2 3
2
4 4 = log 5 3r
5
29. Jika 5 log 3 = a dan 3 log 4 = b maka 4 log15 = ....... Jawab : 4
30. Jika
2
log15= 4 log 3+ 4 log 5 =
3
1 1 1 1 a+ 1 + 5 = + = 3 log 4 log 3. log 4 b ab ab
log a + 2 log b = 12 dan 3.2 log a − 2 log b = 4 maka a+ b = ……..
Jawab : 2
log a + 2 log b = 12⇔ 2 log ab = 12 ⇔ ab = 212
a3 a3 3. log a − log b = 4⇔ log = 4 ⇔ b = 4 b 2 3 3 a a Substitusi b = 4 ke ab = 212 ⇒ a. 4 = 212 ⇔ a 4 = (24 ) 4 ⇒ a = 16 2 2 4 3 (2 ) b= = 28 = 256 4 2 a + b = 16 + 256 = 272 2
2
2
31. 4 log(2 x 2 − 4 x + 16)= 2 log( x + 2)
mempunyai penyelesaian p dan q. Untuk p > q maka nilai
p – q = …….. Jawab :
log(2 x 2 − 4 x + 16)= 4 log( x + 2) 2 ⇒ ( x − 2)( x − 6) = 0 p− q = 6− 2 = 4
4
32. Jika x1 dan x2
akar-akar persamaan x ( 2 + log x ) = 1000 maka x1 x2 = ........
Jawab :
2 + log x = x log1000 ⇔ 2 + log x =
3 log x
. log x
(log x + 3)(log x − 1) = 0 1 log x1 = − 3 ⇔ x1 = 1000 log x2 = 1 ⇔ x2 = 10 x1 x2 = 10− 2
33. Jika a dan b akar-akar persamaan 3 Jawab :
3
3
log( 4 x 2 + 3)
+ (2 2 )
2
log( x 2 − 1)
3
log( 4 x 2 + 3)
+ 4
2
log( x 2 − 1)
= 39
4 x 2 + 3 + ( x 2 − 1) 2 = 3 ⇔ ( x 2 + 7)( x 2 − 5) = 0 x1 = a = 5 x2 = b = − 5 a+ b = 0
= 39 maka a + b = …….
6
34. Tentukan jumlah dari penyelesaian persamaan 2 log 2 x + 5.2 log x + 6 = 0 Jawab :
(
2
log x + 2
)(
2
)
log x + 3 = 0
2
log x1 = − 2 ⇔ x1 =
1 4
2
log x2 = − 3 ⇔ x2 =
1 8
x1 + x2 =
3 8
35. Tentukan penyelesaian 2 log x + log 6 x − log 2 x − log 27 = 0 Jawab :
log 6 x 3 = log 54 x ⇒ 6 x ( x − 3)( x + 3) = 0 ⇒ x = 3
x 36. Tentukan hasil kali semua nilai x yang memenuhi persamaan log 64 2 24
Jawab :
log 2
6+
x 2 − 40 x 24
= log1 = log 20
x 2 − 40 x + 144 = 0 ⇒ x1 x2 =
144 = 144 1
3
log x 3 maka f ( x) + f ( ) = ........ 3 1 − 2. log x x
37. Jika f ( x) = Jawab :
3 3 3 log 3x log x log x log 3− 3 log x + = + 1 − 2.3 log x 1 − 2.3 log 3x 1 − 2.3 log x 1 − 2(3 log 3− 3 log x) 3
=
38. Jika
[
a
log x 1− 3 log x 2.3 log x − 1 + = = −1 1 − 2.3 log x 1 − 2 + 2.3 log x 1 − 2.3 log x 3
](
)
log(3x − 1) 5 log a = 3 maka x = ……
Jawab : 5
log a.a log(3 x − 1) = 3 ⇔ 3 x − 1 = 125 ⇔ x = 42
39. Jika log (y + 2) + 2 log x = 1 maka y = ……. Jawab :
log x 2 ( y + 2) = log10 ⇒ y =
40. Jika 2 x + y = 8 dan log ( x + y ) = Jawab :
log ( x + y ) =
10 − 2 x2
3 log 2.8 log 36 maka x 2 + 3 y = ...... 2
. . log 2.2 log 6 = log 6
3 2 2 3
x + y = 6 dan 2 x + y = 8 maka x = 2 dan y = 4 x 2 + 3 y = 16
2
− 40 x
= 0
7
41. Tentukan nilai x yang memenuhi sistem persamaan : 2 log x − log y = 1 log x + log y = 8 Jawab :
log
x2 x2 = log10 ⇒ y = y 10
log xy = log108 ⇒ xy = 108 Substitusi y = x.
x2 ke xy = 108 10
x2 = 108 ⇒ x = 1000 10
42. Jika x1 dan x2
memenuhi persamaan ( 2 log x − 1). x
1 = log10 maka x1 x2 = ....... log10
Jawab :
2 log x − 1 = x1 x2 = g
−
b a
1 ⇔ 2 log 2 x − log x − 1 = 0 log x 1
= 10 2 =
10
43. Tentukan nilai x yang memenuhi persamaan
log x = 4 log(a + b) + 2 log(a − b) − 3 log(a 2 − b 2 ) − log Jawab :
a+ b a− b
(a + b) 4 .( a − b) 2 ( a + b) 4 ( a − b) 2 log x = log ⇔ x= =1 (a 2 − b 2 )3 . aa +− bb ( a + b) 4 ( a − b) 2 44. Jika x1 dan x2
memenuhi persamaan (1 + 2 log x ) log x = log10 maka x1 x2 = ........
Jawab :
2 log 2 x + log x − 1 = 0 ⇒ x1 x2 = 10
45. Tentukan penyelesaian persamaan Jawab :
( ( 2
2 2
) log x )
2
1 2
1 10
=
)
2 2 log x + 2.2 log( ) = 1 x
2
log x + 2( 2 log 2− 2 log x) = 1 2
− 2.2 log x + 1 = 0 ⇔
(
2
)
2
log x − 1 = 0
log x = 1 ⇔ x = 2
46. Jika x1 dan x2 Jawab :
(
−
(
akar-akar persamaan log ( 2 x 2 − 11x + 12) = 1 maka x1 x2 = .......
)
log 2 x 2 − 11x + 12 = log10 ⇒ 2 x 2 − 11x + 2 = 0 ⇒ x1 x2 =
c 2 = =1 a 2
8
penyelesaian persamaan 106 log x − 4(103 log x ) = 12
47. Tentukan Jawab :
(10
)
(
3 log x 2
)
(
)(
)
− 4 103 log x − 12 = 0 ⇔ 103 log x − 6 103 log x + 2 = 0
103 log x = 6 ⇔ log x 3 = log 6 ⇒ x =
3
6
103 log x = − 2 tidak memenuhi
(
)
48. Tentukan penyelesaian persamaan 2 log 2 log 2 x + 1 + 3 = 1+ 2 log x Jawab : 2 2
(
)
log 2 log 2 x + 1 + 3 = 2 log 2 x log(2
(2 )
x 2
x+ 1
+ 3) = 2 x ⇔ 2 x + 1 + 3 = 22 x
(
)(
)
− 2.2 x − 3 = 0 ⇔ 2 x − 3 2 x + 1 = 0
2 x = − 1 tidak memenuhi 2 x = 3 ⇔ x = 2 log 3 49. Jika
( x + 1)
(
)
log x 3 + 3 x 2 + 2 x + 4 = 3 maka x = …….
Jawab :
x 3 + 3 x 2 + 2 x + 4 = ( x + 1)3 x 3 + 3 x 2 + 2 x + 4 = x 3 + 3 x 2 + 3x + 1 ⇔ x = 3 50. Tentukan penyelesaian pertaksamaan 2 log ( x − 2) ≤ log ( 2 x − 1) Jawab :
log ( x − 2 ) ≤ log ( 2 x − 1) 2
x 2 − 6 x + 5 ≤ 0 ⇔ 1 ≤ x ≤ 5 .........(1) Syarat : i. x − 2 > 0 ⇔ x > 2 ...........(2) 1 ii. 2 x − 1 > 0 ⇔ x > ............(3) 2 (1) ∩ (2) ∩ (3) ⇒ 2 < x ≤ 5 51. Tentukan penyelesaian pertaksamaan log ( x + 3) + 2 log 2 > log x 2 Jawab :
log ( 4 x + 12) > log x 2 ⇒ x 2 − 4 x − 12 < 0 ⇔ − 2 < x < 6 .........(1) Syarat : i. x + 3 > 0 ⇔ x > − 3 ............(2) ii. x 2 > 0 ⇔ x < 0 atau x > 0 ..............(3) (1) ∩ (2) ∩ (3) ⇒ − 2 < x < 0 atau 0 < x < 6
9
52. Tentukan penyelesaian pertaksamaan 2 log x − x log 2 > 0 Jawab : 2
2
log x >
2
1 ⇔ log x
log x < − 1 ⇔ x <
(
2
)(
log x − 1
2
)
log x + 1 > 0
1 2
log x > 1 ⇔ x > 2 Karena syarat x > 0 dan x ≠ 1 maka x > 2
2
53. Tentukan penyelesaian pertaksamaan
2
1 1 − 2 <1 log x log x − 1
Jawab : 2
log x − 1− 2 log x 2 log 2 x − 2 log x − 2 2 2 < 0 2 log 2 x − 2 log x log x − log x
2
log 2 x − 2 log x + 1 > 0 2 log 2 x − 2 log x
Karena 2 log 2 x − 2 log x + 1 definit positif maka : log 2 x − 2 log x > 0⇔ 2 log x < 0 atau 2 log x > 1 ⇔ x < 1 atau x > 2 Karena syarat x > 0 maka : 0 < x < 1 atau x > 2
2
54. Tentukan penyelesaian pertaksamaan 2 log (1 − 2 x ) < 3 1
Jawab : 1 1 1 7 log (1 − 2 x ) < 2 log ⇒ 1 − 2 x > ⇔ x < .........(1) 8 8 16 1 Syarat : 1 − 2 x > 0 ⇔ x < ...........(2) 2 7 (1) ∩ (2) ⇒ x < 16 1 2
1
(
)
55. Tentukan penyelesaian pertaksamaan 2 log x 2 − 3 > 0 Jawab : 1 2
(
)
1
log x 2 − 3 > 2 log1 ⇒ − 2 < x < 2 ............(1)
Syarat : x 2 − 3 > 0 ⇔ x < − 3 atau x > (1) ∩ (2) ⇒ − 2 < x < − 3 atau
3 ..........(2)
3< x< 2
