ࡱ>  !M[ Ebjbj .ΐΐ>=@rr8\aT1111J5J5J5[[[[[[[]o`[J54J5J5J5[11[SSSJ511[SJ5[SSrzZTV[1_wNJZ[[0\Zzaa2RfaaV[aaV[XJ5J5SJ5J5J5J5J5[[SJ5J5J5\J5J5J5J5aaJ5J5J5J5J5J5J5J5J5r {: BENTUK PANGKAT, AKAR, DAN LOGARITMA BENTUK PANGKAT 1.1 PANGKAT BULAT POSITIF Jika  EMBED Equation.2  dan  EMBED Equation.2  maka didefinisikan :  EMBED Equation.2  sebanyak n faktor. a disebut bilangan pokok (dasar) dan n disebut eksponen (pangkat) Contoh 1 : Tentukan nilai dari  EMBED Equation.2  dan  EMBED Equation.2  Jawab :  EMBED Equation.2  = ..  EMBED Equation.2  = .. Contoh 2 : Dengan menguraikan menjadi perkalian, tentukan bentuk eksponen yang paling sederhana dari : a)  EMBED Equation.2  c)  EMBED Equation.2  e)  EMBED Equation.2  b)  EMBED Equation.2  d)  EMBED Equation.2  Jawab : a)  EMBED Equation.2  = . b)  EMBED Equation.2  = . c)  EMBED Equation.2  = . d)  EMBED Equation.2  = .. e)  EMBED Equation.2  = Dari contoh 2 di atas dapat disimpulkan : Jika  EMBED Equation.2 ,  EMBED Equation.2  dan  EMBED Equation.2  maka berlaku sifat-sifat eksponen sbb:  1.  EMBED Equation.2  4.  EMBED Equation.2  2.  EMBED Equation.2  5.  EMBED Equation.2  3.  EMBED Equation.2  Contoh 3 : Dengan menggunakan sifat-sifat eksponen di atas, sederhanakan bentuk berikut : a)  EMBED Equation.2  d)  EMBED Equation.2  b)  EMBED Equation.2  e)  EMBED Equation.2  c)  EMBED Equation.2  f)  EMBED Equation.2  Jawab : a)  EMBED Equation.2  = ... b)  EMBED Equation.2  = ... c)  EMBED Equation.2  = .... d)  EMBED Equation.2  = ... e)  EMBED Equation.2  = .... f)  EMBED Equation.2  = ... LATIHAN SOAL 1. Sederhanakan a)  EMBED Equation.2  f)  EMBED Equation.2  k)  EMBED Equation.2  b)  EMBED Equation.2  g)  EMBED Equation.2  l)  EMBED Equation.2  c)  EMBED Equation.2  h)  EMBED Equation.2  m)  EMBED Equation.2  d)  EMBED Equation.2  i)  EMBED Equation.2  n)  EMBED Equation.2  e)  EMBED Equation.2  j)  EMBED Equation.2  o)  EMBED Equation.2  2. Sederhanakan a)  EMBED Equation.2  b)  EMBED Equation.2  c)  EMBED Equation.2  d)  EMBED Equation.2  1.2. PANGKAT BULAT NEGATIF DAN NOL Contoh 1 : Dengan menggunakan sifat-sifat eksponen, tentukan hubungannya dari : a)  EMBED Equation.2  b)  EMBED Equation.2  c)  EMBED Equation.2  d)  EMBED Equation.2  Jawab : a)  EMBED Equation.2  = .. b)  EMBED Equation.2  = c)  EMBED Equation.2  = .. d)  EMBED Equation.2  = . Dari contoh 1 di atas dapat disimpulkan bahwa : Untuk setiap  EMBED Equation.2  dan  EMBED Equation.2  berlaku sifat-sifat :  1.  EMBED Equation.2  2.  EMBED Equation.2  Contoh 2: Sederhanakan dan jadikan pangkat positif dari : a)  EMBED Equation.2  b)  EMBED Equation.2  c)  EMBED Equation.2  Jawab : a)  EMBED Equation.2  = ... b)  EMBED Equation.2  = ... c)  EMBED Equation.2  = ... LATIHAN SOAL 1. Sederhanakan dan nyatakan dengan eksponen positif dari : a)  EMBED Equation.2  f)  EMBED Equation.2  k)  EMBED Equation.2  b)  EMBED Equation.2  g)  EMBED Equation.2  l)  EMBED Equation.2  c)  EMBED Equation.2  h)  EMBED Equation.2  m)  EMBED Equation.2  d)  EMBED Equation.2  i)  EMBED Equation.2  e)  EMBED Equation.2  j)  EMBED Equation.2  2. Jika a = 2, b = 3 dan c = -2. maka tentukan : a)  EMBED Equation.2  b)  EMBED Equation.2  c)  EMBED Equation.2  1.3 EKSPONEN RASIONAL (PECAHAN) Seperti kita ketahui jika  EMBED Equation.2  maka  EMBED Equation.2  Maka jika  EMBED Equation.2  maka 2 = ...  EMBED Equation.2  maka 2 = ...  EMBED Equation.2  maka 3 = ... Misal  EMBED Equation.2 , jika kedua ruas dipangkatkan n, maka :  EMBED Equation.2   EMBED Equation.2   EMBED Equation.2  Jadi :   EMBED Equation.2  sehingga  EMBED Equation.2  Contoh 1: Ubah ke bentuk akar dari : a)  EMBED Equation.2  b)  EMBED Equation.2  c)  EMBED Equation.2  Jawab : a)  EMBED Equation.2  = .... b)  EMBED Equation.2  = .... c)  EMBED Equation.2  = .... Contoh 2: Ubah ke bentuk pangkat dari : a)  EMBED Equation.2  b)  EMBED Equation.2  Jawab : a)  EMBED Equation.2  = ... b)  EMBED Equation.2 = ..... Contoh 3: Tentukan nilai dari  EMBED Equation.2  Jawab :  EMBED Equation.2  =  EMBED Equation.2  = ..... = ......... LATIHAN SOAL 1. Ubah menjadi bentuk akar a)  EMBED Equation.2  b)  EMBED Equation.2  c)  EMBED Equation.2  d)  EMBED Equation.2  e)  EMBED Equation.2  2. Ubah ke bentuk pangkat a)  EMBED Equation.2  b)  EMBED Equation.2  c)  EMBED Equation.2  d)  EMBED Equation.2  e)  EMBED Equation.2  3. Tentukan nilainya a)  EMBED Equation.2  b)  EMBED Equation.2  c)  EMBED Equation.2  d)  EMBED Equation.2  e)  EMBED Equation.2  4. Sederhanakan dalam bentuk akar a)  EMBED Equation.2  b)  EMBED Equation.2  c)  EMBED Equation.2  d)  EMBED Equation.2  e)  EMBED Equation.2  5. Jika a = 1, b = 3 dan c = -18, maka tentukan x dari  EMBED Equation.2  2. BENTUK AKAR 2.1 OPERASI BENTUK AKAR Bentuk akar termasuk bilangan irasional, yaitu bilangan yang tidak dapat dinyatakan dengan pecahan a/b, a dan b bilangan bulat dan  EMBED Equation.2  Contoh bentuk akar :  EMBED Equation.2  dsb bukan bentuk akar :  EMBED Equation.2  dsb Catatan :  EMBED Equation.2  adalah bilangan non negatif, jadi  EMBED Equation.2  Operasi Pada Bentuk Akar  1.  EMBED Equation.2  2.  EMBED Equation.2  3.  EMBED Equation.2  4.  EMBED Equation.2  Contoh 1: Sederhanakan : a)  EMBED Equation.2  b)  EMBED Equation.2  c)  EMBED Equation.2  d)  EMBED Equation.2  Jawab : a)  EMBED Equation.2  = ... b)  EMBED Equation.2  = .... c)  EMBED Equation.2  = .... d)  EMBED Equation.2  = .... Contoh 2: Sederhanakan : a)  EMBED Equation.2  b)  EMBED Equation.2  c)  EMBED Equation.2  Jawab : a)  EMBED Equation.2  = ... b)  EMBED Equation.2  = ... c)  EMBED Equation.2  = .... Contoh 3 : Sederhanakan : a)  EMBED Equation.2  b)  EMBED Equation.2  c)  EMBED Equation.2  Jawab : a)  EMBED Equation.2  = .... b)  EMBED Equation.2  = .... c)  EMBED Equation.2  = .... LATIHAN SOAL 1. Sederhanakan a)  EMBED Equation.2  b)  EMBED Equation.2  c)  EMBED Equation.2  d)  EMBED Equation.2  e)  EMBED Equation.2  f)  EMBED Equation.2  g)  EMBED Equation.2  h)  EMBED Equation.2  i)  EMBED Equation.2  j)  EMBED Equation.2  2. Sederhanakan a)  EMBED Equation.2  d)  EMBED Equation.2  b)  EMBED Equation.2  e)  EMBED Equation.2  c)  EMBED Equation.2  3. Sederhanakan a)  EMBED Equation.2  c)  EMBED Equation.2  b)  EMBED Equation.2  d)  EMBED Equation.2  2.2 MERASIONALKAN PENYEBUT PECAHAN BENTUK AKAR Jika kita menghitung bilangan, operasi perkalian lebih mudah daripada pembagian. Apalagi operasi pembagian dengan bentuk akar. Ada 3 cara merasionalkan penyebut bentuk pecahan bentuk akar, yaitu : 1. Pecahan Bentuk  EMBED Equation.2  Diselesaikan dengan mengalikan  EMBED Equation.2  Contoh 1: Rasionalkan penyebut dari pecahan : a)  EMBED Equation.2  b)  EMBED Equation.2  Jawab : a)  EMBED Equation.2  =  EMBED Equation.2  x ... = ..... b)  EMBED Equation.2  =  EMBED Equation.2  x ... = ..... 2. Pecahan Bentuk  EMBED Equation.2  Diselesaikan dengan mengalikan  EMBED Equation.2  Contoh 2 : Rasionalkan penyebut pecahan  EMBED Equation.2  Jawab :  EMBED Equation.2  =  EMBED Equation.2  x ... = .... 3. Pecahan Bentuk  EMBED Equation.2  Diselesaikan dengan mengalikan  EMBED Equation.2  Contoh 3 : Rasionalkan penyebut dari pecahan  EMBED Equation.2  Jawab :  EMBED Equation.2  =  EMBED Equation.2 x .... = ........ LATIHAN SOAL 1. Rasionalkan penyebutnya a)  EMBED Equation.2  b)  EMBED Equation.2  c)  EMBED Equation.2  d)  EMBED Equation.2  e)  EMBED Equation.2  2. Rasionalkan penyebutnya a)  EMBED Equation.2  b)  EMBED Equation.2  c)  EMBED Equation.2  d)  EMBED Equation.2  e)  EMBED Equation.2  3. Rasionalkan penyebutnya a)  EMBED Equation.2  b)  EMBED Equation.2  c)  EMBED Equation.2  d)  EMBED Equation.2  e)  EMBED Equation.2  3. PERSAMAAN EKSPONEN (SEDERHANA) Persamaan eksponen yaitu persamaan yang eksponen/pangkatnya mengandung variabel/peubah.  1. Jika  EMBED Equation.2  maka f(x) = p 2. Jika  EMBED Equation.2  maka f(x) = g(x) dimana p suatu konstanta Contoh 1: Tentukan HP dari : a)  EMBED Equation.2  b)  EMBED Equation.2  Jawab : a)  EMBED Equation.2  b)  EMBED Equation.2   EMBED Equation.2  .... = ....  EMBED Equation.2  ..... = .... ..........= .... ...... = ..... x = ... x = .... HP:{............} HP:{ ....... } LATIHAN SOAL Tentukan HP dari : 1.  EMBED Equation.2  6.  EMBED Equation.2  2.  EMBED Equation.2  7.  EMBED Equation.2  3.  EMBED Equation.2  8.  EMBED Equation.2  4.  EMBED Equation.2  9.  EMBED Equation.2  5.  EMBED Equation.2  10.  EMBED Equation.2   EMBED Equation.2  3. LOGARITMA PENGERTIAN LOGARITMA Seperti telah kita ketahui bahwa : Jika  EMBED Equation.3  maka 5 = Jika  EMBED Equation.3  maka  EMBED Equation.3  Jika  EMBED Equation.3  maka 2 = Pada  EMBED Equation.3 , bagaimana menyatakan 3 dengan 2 dan 8? Untuk itu diperlukan notasi yang disebut Logaritma untuk menyatakan pangkat dengan bilangan pokok (basis) dengan hasil pangkat (numerus). Jadi jika  EMBED Equation.3  maka  EMBED Equation.3  dibaca 2 log 8 Sehingga logaritma merupakan invers dari perpangkatan. Secara umum dapat dinyatakan :  Jika  EMBED Equation.3 maka x = . syarat :  EMBED Equation.3  a : basis logaritma y : numerus x : hasil logaritma Khusus untuk bilangan pokok 10, bisa dituliskan bisa juga tidak. Jadi jika log 5 maksudnya  EMBED Equation.3 . Contoh 1: Nyatakan dalam bentuk logaritma dari perpangkatan : a.  EMBED Equation.3  b.  EMBED Equation.3  c.  EMBED Equation.3  Jawab : a.  EMBED Equation.3   EMBED Equation.3  4 = . b.  EMBED Equation.3   EMBED Equation.3  n = . c.  EMBED Equation.3   EMBED Equation.3  b = . Contoh 2 : Nyatakan dalam perpangkatan dari bentuk logaritma : a.  EMBED Equation.3  b. log 100 = 2 c.  EMBED Equation.3  Jawab : a.  EMBED Equation.3   EMBED Equation.3  . b. log 100 = 2  EMBED Equation.3  . c.  EMBED Equation.3   EMBED Equation.3  . Contoh 3: Hitunglah : a.  EMBED Equation.3  b.  EMBED Equation.3  c. log 1000 d.  EMBED Equation.3  e.  EMBED Equation.3  f.  EMBED Equation.3  g.  EMBED Equation.3  Jawab : a.  EMBED Equation.3  = x  EMBED Equation.3  = 64  EMBED Equation.3  x = . b.  EMBED Equation.3  = x  EMBED Equation.3  =  EMBED Equation.3  x = . c. log 1000 = x  EMBED Equation.3  =  EMBED Equation.3  x = . d.  EMBED Equation.3  = x  EMBED Equation.3  =  EMBED Equation.3  x = . e.  EMBED Equation.3  = x  EMBED Equation.3  =  EMBED Equation.3  x = . f.  EMBED Equation.3  = x  EMBED Equation.3  =  EMBED Equation.3  x = . g.  EMBED Equation.3  = x  EMBED Equation.3  =  EMBED Equation.3  x = . LATIHAN SOAL Nyatakan dalam bentuk logaritma dari : a.  EMBED Equation.3  b.  EMBED Equation.3  c.  EMBED Equation.3  d.  EMBED Equation.3  e.  EMBED Equation.3  Nyatakan dalam bentuk perpangkatan dari : a. log 10.000 = 4 b.  EMBED Equation.3  c.  EMBED Equation.3  d.  EMBED Equation.3  e.  EMBED Equation.3  Tentukan nilainya dari : a.  EMBED Equation.3  b.  EMBED Equation.3  c.  EMBED Equation.3  d. log 0,1 e.  EMBED Equation.3  f.  EMBED Equation.3  g.  EMBED Equation.3  h.  EMBED Equation.3  i.  EMBED Equation.3  j.  EMBED Equation.3  k.  EMBED Equation.3  l.  EMBED Equation.3  m.  EMBED Equation.3  SIFAT-SIFAT LOGARITMA Jika  EMBED Equation.3 , maka : 1.  EMBED Equation.3  5.  EMBED Equation.3  2.  EMBED Equation.3  6.  EMBED Equation.3  3.  EMBED Equation.3  7.  EMBED Equation.3  4.  EMBED Equation.3  8.  EMBED Equation.3  Bukti : Sifat 1: Misal  EMBED Equation.3   EMBED Equation.3  Maka bc = . = .  EMBED Equation.3  = + Sifat 6: Misal  EMBED Equation.3   EMBED Equation.3  Sifat 8: Misal  EMBED Equation.3   EMBED Equation.3  Contoh 1: Sederhanakan : a.  EMBED Equation.3  b.  EMBED Equation.3  c.  EMBED Equation.3  d.  EMBED Equation.3  e.  EMBED Equation.3  f.  EMBED Equation.3  g.  EMBED Equation.3  Jawab : a.  EMBED Equation.3  = . b.  EMBED Equation.3 = .. c.  EMBED Equation.3 = . d.  EMBED Equation.3 = .. e.  EMBED Equation.3  = .. f.  EMBED Equation.3  = . g.  EMBED Equation.3  = . Contoh 2: Jika log 2 = 0,3010 dan log 3 = 0,4771, maka tentukan log 24 Jawab : log 24 = . Contoh 3: Jika  EMBED Equation.3 , maka tentukan  EMBED Equation.3  Jawab :  EMBED Equation.3 =  EMBED Equation.3  LATIHAN SOAL Sederhanakan a.  EMBED Equation.3  f.  EMBED Equation.3  b.  EMBED Equation.3  g.  EMBED Equation.3  c.  EMBED Equation.3  h.  EMBED Equation.3  d.  EMBED Equation.3  i.  EMBED Equation.3  e.  EMBED Equation.3  j.  EMBED Equation.3  Jika log 2 = 0,3010 dan log 5 = 0,6990, maka tentukan : a. log 20 b. log 500 c. log 40 d.  EMBED Equation.3  e.  EMBED Equation.3  Jika  EMBED Equation.3 , maka tentukan : a.  EMBED Equation.3  b.  EMBED Equation.3  c.  EMBED Equation.3  d.  EMBED Equation.3  e.  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L .1  & & MathType-zOle PIC LMETA CompObjZTimes New Roman- 2 2 Times New Roman- 2 [2p 2 P4pTimes New Roman- 2 p 2 -qSymbol- 2 1 2 71 2 !1 2  2 7 2 ! & "System-n]]] FMicrosoft Equation 2.0 DS Equation Equation.2`+P)'**'* 2p 2 q() 42 -qL!NObjInfoEquation Native |_10208068271FPTw`BTwOle PIC LMETA CompObj ZObjInfo     #%&'()*+,.146789:;<=>?ADGIJKLMNOPRUXZ[\]^_`abdgjlmnopqrstuwz}!N? a .1  `&  & MathTypeSymbol-2 6,(Symbol-2 6~)Times New Romanw- 2 2 Times New RomanDw- 2 T3p 2 4p 2 T2pTimes New Roman- 2 xy 2  x 2 byTimes New Romanw - 2 .` & "System-@  FMicrosoft Equation 2.0 DS Equation Equation.2Equation Native |_1434225651JF`BTw`BTwOle PIC L`+\)'*L0'* 2xy 3 () 4 .x 2 y2pLE{lhE{^?  .1  @& & MathTypePTimes New Roman-META CompObjfObjInfoEquation Native H 2 `p 2 xp Times New RomanF- 2 06p 2 )4p & "System-K*L8%8 FMicrosoft Equation 3.0 DS Equation Equation.39qV, 4D p 6 p 4L"X,"?  .1  & & MathType Times New Roman- 2 @x 2 Yx Time_1020806997F`BTw`BTwOle !PIC "LMETA $s New Roman- 2 10pp 2 3pTimes New Roman- 2 :i & "System- FMicrosoft Equation 2.0 DS Equation Equation.2CompObj-ZObjInfo/Equation Native 0\_1020807163#F`BTwpiTw@+4*'*(1'* x 10 :x 3 L<N<N> @ .1  & & MathTypeSymbol-2 6,Ole 2PIC 3LMETA 5CompObj@Z(Symbol-2 6)Symbol- 2 -Times New Roman- 2 3 Times New Roman- 2 T3p 2 +3pTimes New Roman- 2 :k & "System- FMicrosoft Equation 2.0 DS Equation Equation.2@+*'*1'* -3k 3 () 3Lp,ObjInfoBEquation Native C\_1434225666FpiTwpiTwOle EPIC FLMETA HCompObjQfObjInfoS^?  .1  &@ & MathType Times New Roman- 2 @4 2  2 Times New Roman&- 2 3pTimes New Roman- 2 a 2 bx 2 a & "System- FMicrosoft Equation 3.0 DS Equation Equation.39qV* TE 4a2a 3L ,Equation Native TF_1020807026FpiTwpiTwOle VPIC WL?  .1  & & MathType Times New Roman- 2 08 2 2 Times New Roman- 2 5p 2 C2pTimes New Roman- 2 k 2 ckTimes NeMETA YHCompObjcZObjInfoeEquation Native f\w Roman- 2 7:i & "System-TTTT FMicrosoft Equation 2.0 DS Equation Equation.2@+X*'*L1'* 8k 5 :2k 2_1020807187FpiTwTwOle hPIC iLMETA kL_N _N> ( .1  & & MathTypeSymbol-2 6(Symbol-2 6)Times New Roman- 2 @2 2 {5 Times New Roman- 2 T2p 2 <3pTimes New Roman- 2 Ip & "System-ooooww FMicrosoft Equation 2.0 DS Equation Equation.2CompObjvZObjInfoxEquation Native y\_1434225648 FTwTw@+ +'*1'* 25p 2 () 3L{`h{0  .1  @& &Ole {PIC  |LMETA ~(CompObj  f MathTypePTimes New Roman- 2 @2 2 6 Times New Roman&- 2 2p 2 S5pTimes New Roman- 2 .p 2 xpx 2 p & "System-  7?  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