Sabtu, 06 November 2010

Matematika (Logaritma & Persamaan Linier)


LOGARITMA


  1. Jika log 5 = 0,699 dan log 3 = 0,477 , maka log 150 = ………….
Jawab : log 150 = log (5 x 3 x 10)
                          = log 5 + log 3 + log 10
                          = 0,699 + 0,477 + 1
                          = 2,176

  1. Nilai dari 5 log 10 + 5 log 50 – 5 log 4 = …………
Jawab : 5 log (10 x 50)
                             4
             5 log (500)
                           4
             5 log 125 = y
             125          = 5 y
              5 ³           = 5 y
              y             = 3
              
  1. Nilai x yang memenuhi persamaan 2 log (x – 2) + 2 log (x – 3) = 2 log 3 . 3 log 2 adalah …………
Jawab : 2 log (x – 2) (x – 3) = 2 log 2
                      (x – 2) (x – 3) = 2
                      x ² - 5 x + 6 – 2 = 0
                      (x – 1) = 0   (x – 4) = 0
                       x – 1 = 0      x – 4 = 0
                         x = 1     /     x = 4        

  1. Jika a log 81 – 2a log 27 + a log 243 = 6, maka nilai a = …………..
Jawab : a log 81 – 2a log 27 + a log 243 = 6
             a log 81 – a log 27 ² + a log 243 = 6
             a log (81 . 243) = 6
                      27²
             a log (3⁴   x 3⁵)  = 6
                        3³·² 
             a log (3 ⁽⁴⁻⁶⁺⁵⁾)  = 6
             a log 3³             = 6
                                 a⁶     = 3³
                               a      = 3½     

  1. Jika log 2 = p dan log 3 = q, maka log 12 = ………….
Jawab : log 12 = log (2² x 3)
                         = log 2² + log 3
                         = 2p     +     q


  1. Nilai dari 2 log16 – 3 log 27 + 5 log 1 = …………………
Jawab : 2 log 16 – 3 log 27 + 5 log 1
              =   4       -     3          +      0
             =       1

  1. Nilai x dari log (x + 1) = 1 + log x adalah …………….
Jawab : log (x + 1) = 1 + log x
             log (x + 1) = log 10 + log x
             log (x + 1) = log (10 . x)
                     x + 1  =  10 x
                 x – 10 x =  -1
                    -9 x     =  -1
                          x     =   1/9

  1. Nilai dari log 2 √2 + log √3 + log 18 = ……………………
                                   log 6
Jawab : log 2 √2 + log √3 + log 18
                             log 6
            = log (2 √2 . √3 . 18)
                        log 6
            = log (36 √6)
                    log 6
            = log (6² . 6½)
         log 6
= log (6²⁺¹ ²)
        log 6
= log 6⁵ ²
     log 6
= 5/2 log 6
      log 6
= 5/2

  1. Jika log x² - log x + log √x = 6, maka x = ………………….
Jawab : log x² - log x + log √x = 6
                 log (x² . x ¹ ²)           = 6
                              x
                 log (x ⁽²⁺¹ ²⁻¹⁾)          = 6
                 log . x³ ²                   = 6
                    10⁶ . x³ ²               = 0     
                                     x           = 10 . ⁶/₃ ₂
                                    x           = 10⁶ · ² ³
                                    x           = 10⁴
                                    x           = 10.000 




  1. Nilai dari 2 log 3 . 3 log 5 . 5 log 6 . 6 log 8 = …………....
Jawab : 2 log 3 . 3 log 5 . 5 log 6 . 6 log 8
            = 2 log 5 . 5 log 8
            = 2 log 8
            = 3

  1. Jika 4 log 5 = a, maka 16 log √5 = ………….
Jawab : 16 log √5
            = 4² log 5¹ ²
            = ½   .  4 log 5
                 2
            = ½ . ½ . a
            = ¼ a

  1. Nilai 2 log 40 – 2 log 5 + 2 log 4 = …………
Jawab : 2 log 40 – 2 log 5 + 2 log 4
            = 2 log (40 . 4)
                             5
            = 2 log 160
                          5
            = 2 log 32
            = 5
PERSAMAAN LINIER


  1. Tentukan persamaan garis yang melalui titik P (5, -3) dan Q (-1, -4)
Jawab : m = y₂ - y₁
                     x₂ - x₁
             m = -4 – (-3) = -1 = 1/6
                     -1 – 5        -6
             y – y₁ = m (x - x₁)
             y – (-4) = 1/6 (x – (-1))
             y = 1/6x + 1/6 - 4
             y = 1/6x – 23/6
             6y = x – 23
            x = 6y + 23       y = -23/6
            x = 23              y = -3,83

  1. Harga 5 baju Rp. 50.000,- dan harga 3 baju Rp. 25.000,-. Berapakah harga 10 baju ?
Jawab : x  = baju         y  = harga
 x₁ = 5              y₁ = 50.000
 x₂ = 3              y₂ = 25.000

m = - 25.000 = 12.500
                                    - 2
                        y - y₁ = m (x - x₁)
                        y – 50.000 = 12.500 (x – 5)
                        y = 12.500x – 62.500 + 50.000
                        y = 12.500x – 12.500
                        10 baju = 12.500 (10) – 12.500
                                     = 125.000 – 12.500
                                     = 112.500

  1. Tentukan persamaan garis yang melalui titik A (5, -4) dan B (-5, 1)
Jawab : m = y₂ - y₁
                     x₂ - x₁
             m = 1 + 4  = 5   = -1/2
                     -5 - 5   -10
             y – y₁ = m (x - x₁)
             y – (-4) = -1/2 (x - 5)
             y = -1/2x + 5/2 - 4
             y = -1/2x – 3/2
             2y = -x – 3
             x = -2y – 3      y = -3/2
             x = -3              y = -1,5           

  1. Harga 15 buku Rp. 35.000,- dan harga 8 buku Rp. 14.000,-. Berapakah harga 13 baju ?
Jawab : x  = buku         y  = harga
 x₁ = 15            y₁ = 35.000
 x₂ = 8              y₂ = 14.000

m = - 21.000 = 3.000
                                    - 7
                        y - y₁ = m (x - x₁)
                        y – 35.000 = 3.000 (x – 15)
                        y = 3.000x – 45.000 + 35.000
                        y = 3.000x – 10.000
                        8 buku = 3.000 (13) – 10.000
                                     = 39.000 – 10.000
                                     = 29.000

  1. Tentukan persamaan garis yang melalui titik R (-2, -3) dan S (-3, 6)
Jawab : m = y₂ - y₁
                     x₂ - x₁
             m = 6 + 3  = 9  = -9
                     -3 + 2   -1
             y – y₁ = m (x - x₁)
             y – (-3) = -9 (x – (-2))
             y = -9x - 18 - 3
             y = -9x - 21
            9x + 21 = 0
            9x = -21
              x = -21/9
              x =  -2,3         y = -21     

  1. Harga 2 kg telur Rp. 45.000,- dan harga 6 kg telur Rp. 105.000,-. Berapakah harga 1 kg telur ?
Jawab : x  = telur         y  = harga
 x₁ = 2             y₁ = 45.000
 x₂ = 6              y₂ = 105.000

m = 60.000 = 15.000
                                    4
                        y - y₁ = m (x - x₁)
                        y – 45.000 = 15.000 (x – 2)
                        y = 15.000x – 30.000 + 45.000
                        y = 15.000x + 15.000
                        1 kg telur = 15.000 (1) + 15.000
                                         = 30.000

  1. Tentukan persamaan garis yang melalui titik K (-6, 1) dan L (5, 2)
Jawab : m = y₂ - y₁
                     x₂ - x₁
             m = 2 - 1  = 1  
                     5 + 6    11
             y – y₁ = m (x - x₁)
             y – 1 = 1/11 (x + 6)
             y = 1/11x + 6/11 + 1
             y = 1/11x + 17/11
            11y = x + 17
            x = 11y - 17                 y = 17/11
            x = -17                         y = 1,55
 
  1. Harga 7 penghapus Rp. 14.000,- dan harga 13 buku Rp. 38.000,-. Berapakah harga 25 penghapus ?
Jawab : x  = penghapus           y  = harga
 x₁ = 7                          y₁ = 14.000
 x₂ = 13                                    y₂ = 38.000

m = 24.000 = 4.000
                                    6
                        y - y₁ = m (x - x₁)
                        y – 14.000 = 4.000 (x – 7)
                        y = 4.000x – 28.000 + 14.000
                        y = 4.000x – 14.000
                        25 penghapus = 4.000 (25) – 14.000
                                                = 100.000 – 14.000
                                                = 86.000

  1. Tentukan persamaan garis yang melalui titik M (4, -2) dan N (-2, 8)
Jawab : m = y₂ - y₁
                     x₂ - x₁
             m = 8 + 2  = 10    
                     -2 - 4     -6
             y – y₁ = m (x - x₁)
             y + 2) = -10/6 (x – 4)
             y = -10/6x + 40/6 - 2
             y = -10/6x + 28/6
            6y = -10x + 28
            10x – 28 = 0
            10x = 28
                x = 28/10                 y = 28/6
                x = 0,28                   y = 4,67

  1. Harga 17 penggaris Rp. 32.000,- dan harga 9 penggaris Rp. 20.000,-. Berapakah harga 5 penggaris ?
Jawab : x  = penggaris                        y  = harga
 x₁ = 17                                    y₁ = 32.000
 x₂ = 9                                      y₂ = 20.000

m = - 12.000 = 1.500
                                    - 8
                        y - y₁ = m (x - x₁)
                        y – 32.000 = 1.500 (x – 17)
                        y = 1.500x – 25.500 + 32.000
                        y = 1.500x + 6.500
                        8 buku = 1.500 (5) + 6.500
                                     = 7.500 + 6.500
                                     = 14.000

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