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. ${b^n}^{}log{a}^m$ = m/n ^blog a
11. log a^{log b} = log b^{log a} = log b x log a
12. a^{log b} = b^{log a}
13. ^alog 1 = 0
14. ^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