Soal-Soal Matematika/Eksponen dan logaritma

Bentuk dan sifat eksponen

sunting

bentuk: ab = c

sifat:

  1. amxan = am+n
  2. am:an = am-n
  3. (am)n = amxn
  4.   = am/n
  5. amxbm = (axb)m
  6. am:bm = (a:b)m
  7. a-m =  
  8. amn = a(mn)
  9. (am)n ≠ amn
  10. 0a = 0 dengan a > 0
  11. a0 = 1 dengan a ≠ 0
  12. 0:a = 0 dengan a ≠ 0
  13. a1 = a
  14. a-1 =   dengan a ≠ 0
  15. aa = bb dimana a = b
Tambahan
xx……n = n maka   dengan x ≥ 1
xxn = n maka x = n dengan x ≥ 1
Hasil x dari kedua persamaan yaitu x2 = 4 dan x =   adalah berbeda. Untuk x2 = 4 maka hasilnya x =   sedangkan x =   hasilnya x = 2.

Bentuk dan sifat logaritma

sunting

bentuk: alog c = b dengan {a,c} > 0 dan a ≠ 1

sifat:

  1. log a+log b = log (axb)
  2. log a-log b = log (a:b)
  3. log a:log b = blog a
  4. blog a x alog c = blog c
  5. blog a = log a:log b
  6. blog a = 1:alog b
  7.   = a
  8. blog am = m x blog a
  9.   = 1/n blog a
  10.   = m/n blog a
  11. alog 1 = 0
  12. alog a = 1

contoh soal

Berapa nilai x dari

  1. x2x6 = 36
  2. xx2 = 16
  3. 6x + x = 219
  4. 2x + 2x = 8
  5. 3x + 5x = 101
jawaban
  1. x2x6 = 36
  • x2x6 = 62
  • (x2x6)3 = (62)3
  • (x6)x6 = 66
  • x6 = 6
  •  
  1. xx2 = 16
  • xx2 = 42
  • (xx2)2 = (42)2
  • (x2)x2 = 44
  • x2 = 4
  •  
  •  
cara biasa
  • 6x + x = 219
  • 6x = 219 - x
  • 6219 x 6x = (219 - x) 6219
  • 6219 = (219 - x) 6(219 - x)
  • 63 x 6216 = (219 - x) 6(219 - x)
  • 216 x 6216 = (219 - x) 6(219 - x)
  • 216 = 219 - x
  • x = 3
fungsi lambert (W(a x ea) = a; e = bilangan euler)
  • 6x + x = 219
  • 6x = 219 - x
  • 1 = (219 - x) x 6-x
  • 6219 = (219 - x) x 6-x x 6219
  • 6219 = (219 - x) x 6(219-x)
  • 6219 = (219 - x) x eln 6(219-x)
  • 6219 = (219 - x) x e(219-x)ln 6
  • 6219 x ln 6 = (219 - x) x ln 6 x e(219-x)ln 6
  • W(6219 x ln 6) = W((219 - x) x ln 6 x e(219-x)ln 6)
  • W(6219 x ln 6) = (219 - x) x ln 6
  • W(63 x 6216 x ln 6) = (219 - x) x ln 6
  • W(216 x ln 6 x 6216) = (219 - x) x ln 6
  • W(216 x ln 6 x e(ln 6 x 216)) = (219 - x) x ln 6
  • W(216 x ln 6 x e(216 x ln 6)) = (219 - x) x ln 6
  • 216 x ln 6 = (219 - x) x ln 6
  • 216 = 219 - x
  • x = 3
cara biasa
  • 2x + 2x = 8
  • 2x = 8 - 2x
  • 2x = 2(4 - x)
  • 24 x 2x = 2(4 - x) 24
  •   = (4 - x) 2(4 - x)
  • 23 = (4 - x) 2(4 - x)
  • 2 x 22 = (4 - x) 2(4 - x)
  • 2 = 4 - x
  • x = 2
fungsi lambert
  • 2x + 2x = 8
  • 2x = 8 - 2x
  • 1 = (8 - 2x) x 2-x
  • 24 = 2(4 - x) x 2-x x 24
  • 23 = (4 - x) x 2(4-x)
  • 23 = (4 - x) x eln 2(4-x)
  • 23 = (4 - x) x e(4-x)ln 2
  • 23 x ln 2 = (4 - x) x ln 2 x e(4-x)ln 2
  • W(23 x ln 2) = W((4 - x) x ln 2 x e(4-x)ln 2)
  • W(23 x ln 2) = (4 - x) x ln 2
  • W(2 x 22 x ln 2) = (4 - x) x ln 2
  • W(2 x ln 2 x 22) = (4 - x) x ln 2
  • W(2 x ln 2 x e(ln 2 x 2)) = (4 - x) x ln 2
  • W(2 x ln 2 x e(2 x ln 2)) = (4 - x) x ln 2
  • 2 x ln 2 = (4 - x) x ln 2
  • 2 = 4 - x
  • x = 2
cara biasa
  • 3x + 5x = 101
  • 3x = 101 - 5x
  • 3x = 5(  - x)
  • 3  x 3x = 5(  - x) 3 
  •   = (  - x) 3(  - x)
  •   3  3  = (  - x) 3(  - x)
  •   34 3  = (  - x) 3(  - x)
  •   x 3  = (  - x) 3(  - x)
  •   =   - x
  • x =  
  • x =  
  • x = 4
fungsi lambert
  • 3x + 5x = 101
  • 3x = 101 - 5x
  • 1 = (101 - 5x) x 3-x
  • 3  = 5(  - x) x 3-x x 3 
  •   = (  - x) x 3( -x)
  •   = (  - x) x eln 3( -x)
  •   = (  - x) x e( -x)ln 3
  •   x ln 3 = (  - x) x ln 3 x e( -x)ln 3
  • W(  x ln 3) = W((  - x) x ln 3 x e( -x)ln 3)
  • W(  x ln 3) = (  - x) x ln 3
  • W(  x ln 3) = (  - x) x ln 3
  • W(  x ln 3 x  ) = (  - x) x ln 3
  • W(  x ln 3 x  ) = (  - x) x ln 3
  • W(  x ln 3 x  ) = (  - x) x ln 3
  • W(  x ln 3 x e(ln 3 x  )) = (  - x) x ln 3
  • W(  x ln 3 x e(  x ln 3)) = (  - x) x ln 3
  •   x ln 3 = (  - x) x ln 3
  •   =   - x
  • x =   -  
  • x =  
  • x = 4