Soal-Soal Matematika/Eksponen dan logaritma

Bentuk dan sifat eksponen

bentuk: a^b = c

sifat:

1. a^mxaⁿ = a^m+n
2. a^m:aⁿ = a^m-n
3. (a^m)ⁿ = a^mxn
4. ${\sqrt[{n}]{a^{m}}}$ = a^m/n
5. a^mxb^m = (axb)^m
6. a^m:b^m = (a:b)^m
7. a^-m = ${\frac {1}{a^{m}}}$
8. a^mn = a^(mn)
9. (a^m)ⁿ ≠ a^mn
10. 0^a = 0 dengan a > 0
11. a⁰ = 1 dengan a ≠ 0
12. 0:a = 0 dengan a ≠ 0
13. a¹ = a
14. a^-1 = ${\frac {1}{a}}$ dengan a ≠ 0
15. a^a = b^b dimana a = b

Tambahan

x^x……n = n maka $x={\sqrt[{n}]{n}}$ dengan x ≥ 1

x^xn = n maka x = n dengan x ≥ 1

Hasil x dari kedua persamaan yaitu x² = 4 dan x = ${\sqrt {4}}$ adalah berbeda. Untuk x² = 4 maka hasilnya x = $\pm {2}$ sedangkan x = ${\sqrt {4}}$ hasilnya x = 2.

Bentuk dan sifat logaritma

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. $b^{^{b}loga}$ = a
8. ^blog a^m = m x ^blog a
9. $^{b^{n}}loga$ = 1/n ^blog a
10. $^{b^{n}}loga^{m}$ = m/n ^blog a
11. ^alog 1 = 0
12. ^alog a = 1

contoh soal

Berapa nilai x dari

1. x^2x6 = 36
2. x^x2 = 16
3. 6^x + x = 219
4. 2^x + 2x = 8
5. 3^x + 5x = 101

jawaban

1. x^2x6 = 36

• x^2x6 = 6²
• (x^2x6)³ = (6²)³
• (x⁶)^x6 = 6⁶
• x⁶ = 6
• $x={\sqrt[{6}]{6}}$

1. x^x2 = 16

• x^x2 = 4²
• (x^x2)² = (4²)²
• (x²)^x2 = 4⁴
• x² = 4
• $x={\sqrt {4}}$
• $x=2$

cara biasa

• 6^x + x = 219
• 6^x = 219 - x
• 6²¹⁹ x 6^x = (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 e^a) = a; e = bilangan euler)

• 6^x + x = 219
• 6^x = 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 e^{ln 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

• 2^x + 2x = 8
• 2^x = 8 - 2x
• 2^x = 2(4 - x)
• 2⁴ x 2^x = 2(4 - x) 2⁴
• ${\frac {2^{4}}{2}}$ = (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

• 2^x + 2x = 8
• 2^x = 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 e^{ln 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

• 3^x + 5x = 101
• 3^x = 101 - 5x
• 3^x = 5(${\frac {101}{5}}$ - x)
• 3^{${\frac {101}{5}}$} x 3^x = 5(${\frac {101}{5}}$ - x) 3^{${\frac {101}{5}}$}
• ${\frac {3^{\frac {101}{5}}}{5}}$ = (${\frac {101}{5}}$ - x) 3^{(${\frac {101}{5}}$ - x)}
• ${\frac {1}{5}}$ 3^{${\frac {20}{5}}$} 3^{${\frac {81}{5}}$} = (${\frac {101}{5}}$ - x) 3^{(${\frac {101}{5}}$ - x)}
• ${\frac {1}{5}}$ 3⁴ 3^{${\frac {81}{5}}$} = (${\frac {101}{5}}$ - x) 3^{(${\frac {101}{5}}$ - x)}
• ${\frac {81}{5}}$ x 3^{${\frac {81}{5}}$} = (${\frac {101}{5}}$ - x) 3^{(${\frac {101}{5}}$ - x)}
• ${\frac {81}{5}}$ = ${\frac {101}{5}}$ - x
• x = ${\frac {101}{5}}-{\frac {81}{5}}$
• x = ${\frac {20}{5}}$
• x = 4

fungsi lambert

• 3^x + 5x = 101
• 3^x = 101 - 5x
• 1 = (101 - 5x) x 3^-x
• 3^{${\frac {101}{5}}$} = 5(${\frac {101}{5}}$ - x) x 3^-x x 3^{${\frac {101}{5}}$}
• ${\frac {3^{\frac {101}{5}}}{5}}$ = (${\frac {101}{5}}$ - x) x 3^{(${\frac {101}{5}}$-x)}
• ${\frac {3^{\frac {101}{5}}}{5}}$ = (${\frac {101}{5}}$ - x) x e^{ln 3(${\frac {101}{5}}$-x)}
• ${\frac {3^{\frac {101}{5}}}{5}}$ = (${\frac {101}{5}}$ - x) x e^{(${\frac {101}{5}}$-x)ln 3}
• ${\frac {3^{\frac {101}{5}}}{5}}$ x ln 3 = (${\frac {101}{5}}$ - x) x ln 3 x e^{(${\frac {101}{5}}$-x)ln 3}
• W(${\frac {3^{\frac {101}{5}}}{5}}$ x ln 3) = W((${\frac {101}{5}}$ - x) x ln 3 x e^{(${\frac {101}{5}}$-x)ln 3})
• W(${\frac {3^{\frac {101}{5}}}{5}}$ x ln 3) = (${\frac {101}{5}}$ - x) x ln 3
• W(${\frac {3^{\frac {20}{5}}\cdot 3^{\frac {81}{5}}}{5}}$ x ln 3) = (${\frac {101}{5}}$ - x) x ln 3
• W(${\frac {3^{\frac {20}{5}}}{5}}$ x ln 3 x $3^{\frac {81}{5}}$) = (${\frac {101}{5}}$ - x) x ln 3
• W(${\frac {3^{4}}{5}}$ x ln 3 x $3^{\frac {81}{5}}$) = (${\frac {101}{5}}$ - x) x ln 3
• W(${\frac {81}{5}}$ x ln 3 x $3^{\frac {81}{5}}$) = (${\frac {101}{5}}$ - x) x ln 3
• W(${\frac {81}{5}}$ x ln 3 x e^{(ln 3 x ${\frac {81}{5}}$)}) = (${\frac {101}{5}}$ - x) x ln 3
• W(${\frac {81}{5}}$ x ln 3 x e^{(${\frac {81}{5}}$ x ln 3)}) = (${\frac {101}{5}}$ - x) x ln 3
• ${\frac {81}{5}}$ x ln 3 = (${\frac {101}{5}}$ - x) x ln 3
• ${\frac {81}{5}}$ = ${\frac {101}{5}}$ - x
• x = ${\frac {101}{5}}$ - ${\frac {81}{5}}$
• x = ${\frac {20}{5}}$
• x = 4