Soal-Soal Matematika/Eksponen dan logaritma
Bentuk dan sifat eksponen
suntingbentuk: ab = c
sifat:
- amxan = am+n
- am:an = am-n
- (am)n = amxn
- = am/n
- amxbm = (axb)m
- am:bm = (a:b)m
- a-m =
- amn = a(mn)
- (am)n ≠ amn
- 0a = 0 dengan a > 0
- a0 = 1 dengan a ≠ 0
- 0:a = 0 dengan a ≠ 0
- a1 = a
- a-1 = dengan a ≠ 0
- 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
suntingbentuk: alog c = b dengan {a,c} > 0 dan a ≠ 1
sifat:
- log a+log b = log (axb)
- log a-log b = log (a:b)
- log a:log b = blog a
- blog a x alog c = blog c
- blog a = log a:log b
- blog a = 1:alog b
- = a
- blog am = m x blog a
- = 1/n blog a
- = m/n blog a
- alog 1 = 0
- alog a = 1
contoh soal
Berapa nilai x dari
- x2x6 = 36
- xx2 = 16
- 6x + x = 219
- 2x + 2x = 8
- 3x + 5x = 101
- jawaban
- x2x6 = 36
- x2x6 = 62
- (x2x6)3 = (62)3
- (x6)x6 = 66
- x6 = 6
- 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