Logaritma. maka tentukan nilai x yang memenuhi persamaan. log + = + 1 = x x. x Jawab : = b maka tentukan 12. Jawab : Jawab : Jawab :

Logaritma

1.

2log4₊2log12₋2log6₌ ...

Jawab : 3 8 log 6 12 . 4 log 2 2 _{= =}

2.

_Jika

2log7 = a

_{maka tentukan}

8_log49

Jawab : a 3 2 2 3 2 2 2

8_log49 3_{log7 . log7}

= =

=

3.

_Jika

5log3= a dan 3log4= b

_{maka tentukan}

12_log75

Jawab : ) 1 ( 2 1 1 2 4 log 1 1 4 log . 3 log 3 log 2 4 log 3 log 1 4 log 3 log 1 2 12 log 1 12 log 1 2 3 log 5 log 75 log 3 3 5 5 3 3 5 5 3 5 12 2 12 12 b a a b ab a + + = + + + = + + + = + + + = + = + =

4.

_{Tentukan nilai x yang memenuhi persamaan}

(3x+2)log27₌5log3

Jawab : 41 125 2 3 3 log 27 log 5 3 ) 2 3 ( 3 = ⇔ = + ⇒ = + _{x x} x

5.

_Jika

x≠ 1dan x > 0,

_{maka tentukan nilai x yang memenuhi persamaan}

0 1 4 log . 3 ) 12 log( + − x + = x _x Jawab : 4 16 0 ) 4 )( 16 ( 0 64 12 64 log ) 12 log( 4 log log ) 12 log( 2 2 3 = − = = − + ⇔ = − + = + ⇔ = + + x memenuhi tidak x x x x x x x x x x x x x x

6.

_Jika

x₁ dan x₂

memenuhi persamaan

(4− logx)logx = log1000

_{maka tentukan}

x₁x₂ Jawab : 000 . 10 10 0 3 log 4 log 0 ) log ( ) log ( 1 4 2 1 2 2 1 2 = = ⇒ = − + − = ⇒ = + + − − − x x x x p x x c x b x a a b p p

7.

_{Tentukan penyelesaian pertaksamaan}

6log(_x2 _{− x})_< 1

Jawab : 3 1 0 2 ) 2 ( ) 1 ( ) 2 ..( ... 1 0 0 ) 1 ( 0 : ) 1 ...( 3 2 0 ) 2 )( 3 ( 6 log ) log( 2 6 2 6 < < < < − ⇒ ∩ > < ⇔ > − ⇔ > − < < − ⇔ < + − ⇒ < − x atau x x atau x x x x x Syarat x x x x x

8.

log1. log 1₂. log 1₃ = ... a c b c b a Jawab : 6 log . log . log ). 3 ).( 2 .( 1 log . log . log_b−1b _c−2c _a−3 _{= − − −} a _bb _cc _{a= −} a

9.

alog3 _a.alog_{a a =} ... Jawab : 2 1 2 3 3 1_{. . log . log} log . log 2 3 3 1 = = a a a a a a a a

10.

_Jika

2log3= a

_{maka tentukan}

[ ]

( )

2 1 3 2 − a Jawab :

( )

[ ]

[ ]

₅₁₂1 ) 2 ( 2 3 log 2 1 2 1 6 3 3 2 3 2 = = = ⇔ = − − a a a

11.

_{Jika a = 0,16666…….. maka tentukan}

alog36

Jawab : 100a = 16,666….. 10a = 1,666….. -90a = 15 6 1 = ⇔ a 2 36 log 36 log 6 1 − = = a

12.

_{Jika a = 0,111….. dan b = 0,333….. maka tentukan}

a_logb Jawab : 2 1 3 1 3 1 9 1 log log ... 333 , 0 ... 111 , 0 9 1 = = = = = = b b a a 13.

_Jika

log ₂ 12 2 = b a

maka tentukan

log3

a b Jawab : 2 6 . 3 1 log 3 1 log 3 1 log 3 1 log 6 log 12 log 2 12 log 1 3 2 2 − = − = − =       = = = ⇔ = ⇔ = − b a b a a b a b b a b a b a 14.

_Jika

, 1 1 ... log log log log 2 3 3 2 = > > = = n m maka b dan a n b a dan m b a Jawab : 2 2 2 2 3 2 3 2 ) 3 log ( 3 log . 3 log 2 log 3 log . 2 log 3 log log log . log log _{= = =} = _aa _bb a b b a n m

15.

_Jika

5 1 =

a

_maka

( )( )( )

22log6 39log5 5 log2 ... = a Jawab :

( )

( )

5 3 ) 2 ).( 5 .( 6 ) 5 ( 3 . 6 5 3 . 6 5 3 . 6 1 1 2 log 5 log 1 2 log 5 log . 2 log 5 log 2 1 5 2 1 3 5 3 2 1 5 1 2 3 = = =     =         − − −

16.

5log 27.9log125₊16log32₌ ...

Jawab : 2 7 4 5 4 9 2 4 5 3 5 2 3 2 3 5 2 3 3

5_log3.2_log5 4_{log2 . . log3. log5 . log2} 2 3 = + = + = +

17.

_Jika

alog3=blog27, a> 0,b> 0, a ≠ 1,b ≠ 1 maka alogb= ...

Jawab : 3 27 log log 3 log 27 log log log log 27 log log 3 log _{= ⇔ = ⇔ b=}3 ₌ a b b a a

18.

2. log4 .3log25 3log10 3log32 ... 2 1 3 _{− + − =} Jawab : 0 1 log 32 . 25 10 . 4 log 2 3 3 2 1 = =

19.

... 15 log 45 log 3 log ) 5 5 log( + + ₌ Jawab : 2 5 15 log 15 log 45 . 3 . 5 5 log ₂5 15 ₌ =

20.

(

) (

)

... 12 log 4 log 36 log 3 2 3 2 3 = − Jawab :

(

) (

)

₈ 12 log . 12 log 2 . 2 12 log . 9 log . 144 log 12 log . 4 log 36 log 4 log 36 log 3 2 1 3 3 2 1 3 3 3 2 1 3 3 3 3 = = = − +

21.

( )

_{( )}

( )

... log log log log 2 = + + xy xy y x x Jawab :

(

)

( )

( )

( )

2 5 log log log . . log 2 5 2 = = xy xy xy xy y x x

22.

_Jika

alog81− 2.alog27+alog27+alog243= 6

_{maka tentukan a !}

Jawab : 3 3 729 6 27 243 . 27 . 81 log 6 6 2 = ⇔ a = = ⇒ a = a

23.

_Jika

9log8₌ 3m

_{maka tentukan}

4_log3

Jawab : m m m m m 4 1 2 1 . 2 1 2 log 1 . 3 log . 3 log 3 log 2 2 log 3 2 log . 3 2 log 3 2 1 2 2 1 2 4 3 3 2 3 3 3 2 2 = = = = = = ⇔ = ⇔ =

24.

_Jika

2log3_{= a dan} 3log5_{= bmaka} 4log45₌ ...

Jawab : ) ( 2 1 ) 5 log . 3 log ( 2 1 3 log 5 log . 2 1 3 log . 2 2 5 log 3 log 45 log 2 2 2 2 2 2 2 3 4 2 2 ab a+ = + = + = + =

25.

_Jika

7log2_{= adan} 2log3_{= bmaka} 6log98₌ ...

Jawab : ) 1 ( 2 1 1 2 1 3 log 1 3 log . 2 log 2 log 2 2 log 3 log 1 3 log 2 log 2 6 log 1 6 log 1 . 2 2 log 7 log 98 log 2 2 7 7 2 2 7 7 2 7 6 2 6 6 + + = + + + = + + + = + + + = + = + = b a a b ab a 26.

_Jika

... ) 2 ( ) 2 ( log 12 4 1 ₂ 2 2 2 ₌ − + = + > > y x y x maka xy y x dan y x Jawab : 2 log 8 16 log ) 2 ( ) 2 ( log 8 ) 2 ( 8 4 4 . 16 ) 2 ( 16 4 4 . 2 2 2 2 2 2 2 2 = = − + = − ⇔ = − + = + ⇔ = + + xy xy y x y x xy y x xy xy y x ii xy y x xy xy y x i

27.

_Diketahui

2log3₌ 1,6_dan 2log5₌ 2,3

_{. Tentukan nilai}

9 125 log 2 Jawab : 7 , 3 6 , 1 . 2 3 , 2 . 3 3 log 2 5 log 3 3 log 5 log 9 125 log 2 3 2 2 2 2 2 _{= − = − = − =}

28.

_Jika

8log5_{= r maka} 5log16₌ ...

Jawab : r r r r 3 4 5 log 4 2 log . 4 16 log 3 5 log 5 log . 5 log 2 5 5 2 2 3 1 8 = = = = ⇔ = ⇔ =

29.

_Jika

5log3_{= a dan} 3log4_{= bmaka} 4log15₌ ... Jawab : ab a ab b 1 1 1 4 log . 3 log 1 4 log 1 5 log 3 log 15 log 4 4 _{3 5 3} 4 _{= + = + = + =} +

30.

_Jika

2log_a+2log_{b =} 12_dan3.2log_a−2log_b= 4

_{maka a+ b = ……..}

Jawab : 272 256 16 256 2 2 ) 2 ( 16 ) 2 ( 2 2 . 2 2 2 4 log 4 log log . 3 2 12 log 12 log log 8 4 3 4 4 4 4 12 4 3 12 4 3 4 3 3 2 2 2 12 2 2 2 = + = + = = = = ⇒ = ⇔ = ⇒ = = = ⇔ = ⇔ = − = ⇔ = ⇔ = + b a b a a a a ab ke a b Substitusi a b b a b a ab ab b a

31. 4log(2_x2₋ 4_x+ 16)₌2log(_x+ 2)

_{mempunyai penyelesaian p dan q. Untuk p > q maka nilai}

p – q = ……..

Jawab : 4 2 6 0 ) 6 )( 2 ( ) 2 log( ) 16 4 2 log( 2 4 2 4 = − = − = − − ⇒ + = + − q p x x x x x

32.

_Jika

x₁ dan x₂

_{akar-akar persamaan}

_x(2+logx) ₌ 1000

_maka

_... 2 1x = x Jawab : 2 2 1 2 2 1 1 10 10 1 log 1000 1 3 log 0 ) 1 )(log 3 (log log . log 3 log 2 1000 log log 2 − = = ⇔ = = ⇔ − = = − + = + ⇔ = + x x x x x x x x x x x x x

33.

_{Jika a dan b akar-akar persamaan}

₃3log(4 2 3) ₄2log( 2 1) ₃₉ = + − + x x

maka a + b = …….

Jawab : 0 5 5 0 ) 5 )( 7 ( 3 ) 1 ( 3 4 39 ) 2 ( 3 2 1 2 2 2 2 2 ) 1 log( 2 ) 3 4 log( 2 2 2 3 = + − = = = = = − + ⇔ = − + + = + − + b a b x a x x x x x x x

34.

_{Tentukan jumlah dari penyelesaian persamaan}

2log2_x+ 5.2log_x+ 6₌ 0 Jawab :

(

) (

)

8 3 3 log 2 log 0 3 log 2 log 2 1 8 1 2 2 2 4 1 1 1 2 2 2 = + = ⇔ − = = ⇔ − = = + + x x x x x x x x

35.

_{Tentukan penyelesaian}

2logx+ log6x− log2x− log27= 0

Jawab : 3 0 ) 3 )( 3 ( 6 54 log 6 log _x3 _{= x⇒ x x− x+ = ⇒ x=}

36.

_{Tentukan hasil kali semua nilai x yang memenuhi persamaan}

log 64242 2 40 0 =     x − x Jawab : 144 1 144 0 144 40 2 log 1 log 2 log 2 1 2 0 6 2₂₄40 = = ⇒ = + − = = − + x x x x x x 37.

_Jika

( ) (3) ... log . 2 1 log ) ( ₃ 3 = + − = x f x f maka x x x f Jawab : 1 log . 2 1 1 log . 2 log . 2 2 1 log 1 log . 2 1 log ) log 3 log ( 2 1 log 3 log log . 2 1 log log . 2 1 log log . 2 1 log 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 − = − − = + − − + − = − − − + − = − + − x x x x x x x x x x x x x x

38.

_Jika

[

alog(3_x− 1)

]

(

5log_a

)

₌ 3

_{maka x = ……}

Jawab : 42 125 1 3 3 ) 1 3 log( . log 5 _aa _{x− = ⇔ x− = ⇔ x=}

39. Jika log (y + 2) + 2 log x = 1 maka y = …….

Jawab : 2 10 10 log ) 2 ( log 2 _{+ = ⇒ = 2 −} x y y x

40.

_Jika

log2. log36 3 ...

2 3 ) ( log 8 2_{x+ y = dan x+ y =} 8 _{maka x}2_{+ y=} Jawab : 16 3 4 2 8 2 6 6 log 6 log . 2 log . . ) ( log 2 2 3 2 2 3 = + = = = + = + = = + y x y dan x maka y x dan y x y x

41. Tentukan nilai x yang memenuhi sistem persamaan :

   = + = − 8 log log 1 log log 2 y x y x Jawab : 1000 10 10 . 10 10 10 10 log log 10 10 log log 8 2 8 2 8 8 2 2 = ⇒ = = = = ⇒ = = ⇒ = x x x xy ke x y Substitusi xy xy x y y x

42.

_Jika

x₁ dan x₂

memenuhi persamaan

(

)

log10 10 log 1 . 1 log 2 x− _x =

_maka

_{x1x2 = ...} Jawab : 10 10 0 1 log log 2 log 1 1 log 2 2 1 2 1 2 = = = = − − ⇔ = − −_ab g x x x x x x

43.

_{Tentukan nilai x yang memenuhi persamaan}

b a b a b a b a b a x − + − − − − + +

= 4log( ) 2log( ) 3log( ) log

log 2 2 Jawab : 1 ) ( ) ( ) ( ) ( . ) ( ) .( ) ( log log _{4 2} 2 4 3 2 2 2 4 = − + − + = ⇔ − − + = − + _{a b a b} b a b a x b a b a b a x b a b a

44.

_Jika

x₁ dan x₂

_{memenuhi persamaan}

(

1+ 2logx

)

logx = log10

_maka

x₁x₂ = ...

Jawab : 10 1 10 0 1 log log 2 2 1 2 1 2_{x+ x− = ⇒ xx =} − ₌

45.

_{Tentukan penyelesaian persamaan}

(

2log

)

2₊ 2.2log(2)₌ 1

x x Jawab :

(

)

(

)

(

)

2 1 log 0 1 log 0 1 log . 2 log 1 ) log 2 log ( 2 log 2 2 2 2 2 2 2 2 2 2 = ⇔ = = − ⇔ = + − = − + x x x x x x x

46.

_Jika

x₁ dan x₂

akar-akar persamaan

log

(

2_x2₋ 11_x+ 12

)

₌ 1

_maka

_... 2 1x = x Jawab :

(

)

1 2 2 0 2 11 2 10 log 12 11 2 log 2_{− + = ⇒} 2_{− + = ⇒ 1 2 = = =} a c x x x x x x

47.

_{Tentukan penyelesaian persamaan}

106logx₋ 4

(

103logx

)

₌ 12 Jawab :

(

) (

)

(

)(

)

memenuhi tidak x x x x x x x x 2 10 6 6 log log 6 10 0 2 10 6 10 0 12 10 4 10 log 3 3 3 log 3 log 3 log 3 log 3 2 log 3 − = = ⇒ = ⇔ = = + − ⇔ = − −

48.

_{Tentukan penyelesaian persamaan}

2_log2_log

(

₂x+1_{+ 3}

)

_{= 1+}2_logx Jawab :

(

)

( )

(

)(

)

3 log 3 2 1 2 0 1 2 3 2 0 3 2 . 2 2 2 3 2 2 ) 3 2 log( 2 log 3 2 log log 2 2 2 1 1 2 2 1 2 2 = ⇔ = − = = + − ⇔ = − − = + ⇔ = + = + + + + x memenuhi tidak x x x x x x x x x x x x

49.

_Jika

(x+1)log

(

_x3₊ 3_x2₊ 2_x+ 4

)

₌ 3

_{maka x = …….}

Jawab : 3 1 3 3 4 2 3 ) 1 ( 4 2 3 2 3 2 3 3 2 3 = ⇔ + + + = + + + + = + + + x x x x x x x x x x x

50.

_{Tentukan penyelesaian pertaksamaan}

2log

(

x− 2

)

≤ log

(

2x−1

)

Jawab :

(

)

(

)

5 2 ) 3 ( ) 2 ( ) 1 ( ) 3 ..( ... 2 1 0 1 2 . ) 2 .( ... 2 0 2 . : ) 1 ...( 5 1 0 5 6 1 2 log 2 log 2 2 ≤ < ⇒ ∩ ∩ > ⇔ > − > ⇔ > − ≤ ≤ ⇔ ≤ + − − ≤ − x x x ii x x i Syarat x x x x x

51.

_{Tentukan penyelesaian pertaksamaan}

_log

(

_{x+ 3}

)

_{+ 2log2> logx}2

Jawab :

(

)

6 0 0 2 ) 3 ( ) 2 ( ) 1 ( ) 3 ....( ... 0 0 0 . ) 2 ..( ... 3 0 3 . : ) 1 ...( 6 2 0 12 4 log 12 4 log 2 2 2 < < < < − ⇒ ∩ ∩ > < ⇔ > − > ⇔ > + < < − ⇔ < − − ⇒ > + x atau x x atau x x ii x x i Syarat x x x x x

52.

_{Tentukan penyelesaian pertaksamaan}

2log_x−xlog2_> 0 Jawab :

(

)(

)

2 1 0 2 1 log 1 log 0 1 log 1 log log 1 log 2 2 1 2 2 2 2 2 > ≠ > > ⇔ > < ⇔ − < > + − ⇔ > x maka x dan x syarat Karena x x x x x x x x

53.

_{Tentukan penyelesaian pertaksamaan}

1

1 log 1 log 1 2 2 _x− _x− < Jawab : 2 1 0 : 0 2 1 1 log 0 log 0 log log : 1 log log 0 log log 1 log log 0 log log log log log log log 1 log 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 > < < > > < ⇔ > < ⇔ > − + − > − + − < − − − − − − x atau x maka x syarat Karena x atau x x atau x x x maka positif definit x x Karena x x x x x x x x x x x x

54.

_{Tentukan penyelesaian pertaksamaan}

2_log

(

_{1 2}

)

₃ 1 < − x Jawab :

(

)

16 7 ) 2 ( ) 1 ( ) 2 .( ... 2 1 0 2 1 : ) 1 ...( 16 7 8 1 2 1 8 1 log 2 1 log 2 1 2 1 < ⇒ ∩ < ⇔ > − < ⇔ > − ⇒ < − x x x Syarat x x x

55.

_{Tentukan penyelesaian pertaksamaan}

2_log

(

2 ₃

)

₀

1 > − x Jawab :

(

)

2 3 3 2 ) 2 ( ) 1 ( ) 2 ( ... 3 3 0 3 : ) 1 ..( ... 2 2 1 log 3 log 2 2 2 1 2 1 < < − < < − ⇒ ∩ > − < ⇔ > − < < − ⇒ > − x atau x x atau x x Syarat x x