Soal-Soal Matematika/Eksponen dan logaritma

Bentuk dan sifat eksponen

bentuk: a^b = c [a=basis (bilangan pokok); b=pangkat; c=hasil pangkat]

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 [a=basis (bilangan pokok); b=numerus; c=hasil logaritma]

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
• $x=\sqrt{4}$
• $x=2$

1. 6^x + x = 219

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

1. 2^x + 2x = 8

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

1. 3^x + 5x = 101

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

contoh soal

1. Tentukan nilai x dari $2^{\frac{2(x-1)}{x-3}}+2^{\frac{x+1}{x-3}}=12$!

Jawaban

$$\begin{array}{cc}{2}^{\frac{2(x-1)}{x-3}}+2^{\frac{x+1}{x-3}}& =12\\ 2^{\frac{2x-2}{x-3}}+2^{\frac{x+1}{x-3}}& =12\\ 2^{\frac{2x-6+4}{x-3}}+2^{\frac{x-3+4}{x-3}}& =12\\ 2^{\frac{2x-6}{x-3}}\cdot 2^{\frac{4}{x-3}}+2^{\frac{x-3}{x-3}}\cdot 2^{\frac{4}{x-3}}& =12\\ 2^{\frac{2(x-3)}{x-3}}\cdot 2^{\frac{4}{x-3}}+2^{\frac{x-3}{x-3}}\cdot 2^{\frac{4}{x-3}}& =12\\ 2^2\cdot 2^{\frac{4}{x-3}}+2^1\cdot 2^{\frac{4}{x-3}}& =12\\ 4\cdot 2^{\frac{4}{x-3}}+2\cdot 2^{\frac{4}{x-3}}& =12\\ (4+2)\cdot 2^{\frac{4}{x-3}}& =12\\ 6\cdot 2^{\frac{4}{x-3}}& =12\\ 2^{\frac{4}{x-3}}& =2\\ \frac{4}{x-3}& =1\\ 4& =x-3\\ x& =7\end{array}$$

1. Tentukan nilai x dari $\sqrt[3]{3x-2}+\sqrt[3]{2x-1}=\sqrt[3]{5x-3}$!

Jawaban

$$\begin{array}{cc}\sqrt[3]{3x-2}+\sqrt[3]{2x-1}& =\sqrt[3]{5x-3}\\ \text{misal }3x-2=a,2x-1=b\text{ maka menjadi }3x-2+2x-1=5x-3=a+b\\ \sqrt[3]{a}+\sqrt[3]{b}& =\sqrt[3]{a+b}\\ (\sqrt[3]{a}+\sqrt[3]{b}{)}^3& =(\sqrt[3]{a+b}{)}^3\\ a+b+3\sqrt[3]{ab}(\sqrt[3]{a}+\sqrt[3]{b})& =a+b\\ 3\sqrt[3]{ab}(\sqrt[3]{a}+\sqrt[3]{b})& =0\\ \sqrt[3]{ab}& =0\\ ab& =0\\ (3x-2)(2x-1)& =0\\ x=\frac{2}{3}& \text{ atau }x=\frac{1}{2}\\ \sqrt[3]{a}+\sqrt[3]{b}& =0\\ \sqrt[3]{a}& =-\sqrt[3]{b}\\ a& =-b\\ 3x-2& =-(2x-1)\\ 3x-2& =-2x+1\\ 5x& =3\\ x& =\frac{3}{5}\end{array}$$

1. Tentukan hasil dari $\sqrt[3]{5+2\sqrt{13}}+\sqrt[3]{5-2\sqrt{13}}$!

Jawaban

$$\begin{array}{c}\sqrt[3]{5+2\sqrt{13}}+\sqrt[3]{5-2\sqrt{13}}\\ \text{misal }\sqrt[3]{5+2\sqrt{13}}=a\text{ dan }\sqrt[3]{5-2\sqrt{13}}=b\text{ maka menjadi }a+b=p\\ a+b& =p\\ (a+b{)}^3& =p^3\\ a^3+b^3+3ab(a+b)& =p^3\\ a^3+b^3& =(\sqrt[3]{5+2\sqrt{13}}{)}^3+(\sqrt[3]{5-2\sqrt{13}}{)}^3\\ & =5+2\sqrt{13}+5-2\sqrt{13}\\ & =10\\ ab& =(\sqrt[3]{5+2\sqrt{13}})\cdot (\sqrt[3]{5-2\sqrt{13}})\\ & =\sqrt[3]{(5+2\sqrt{13})(5-2\sqrt{13})}\\ & =\sqrt[3]{25-52}\\ & =\sqrt[3]{-27}\\ & =-3\\ a^3+b^3+3ab(a+b)& =p^3\\ 10+3(-3)p& =p^3\\ 10-9p& =p^3\\ p^3+9p-10& =0\\ (p-1)(p^2+p+10)& =0\\ p=1& \text{ atau (definit positif) }\end{array}$$

1. Tentukan hasil terbesar dari $\sqrt[4]{161+72\sqrt{5}}-\sqrt[4]{161+72\sqrt{5}}$!

Jawaban

$$\begin{array}{c}\sqrt[4]{161+72\sqrt{5}}-\sqrt[4]{161+72\sqrt{5}}\\ \sqrt[2]{\sqrt[2]{161+72\sqrt{5}}}-\sqrt[2]{\sqrt[2]{161+72\sqrt{5}}}\\ \sqrt[2]{\sqrt[2]{161+2\sqrt{6480}}}-\sqrt[2]{\sqrt[2]{161+2\sqrt{6480}}}\\ \sqrt[2]{\sqrt[2]{81+80+2\sqrt{6480}}}-\sqrt[2]{\sqrt[2]{81+80+2\sqrt{6480}}}\\ \sqrt[2]{9+4\sqrt{5}}-\sqrt[2]{9-4\sqrt{5}}\\ \sqrt[2]{9+2\sqrt{20}}-\sqrt[2]{9-2\sqrt{20}}\\ \sqrt[2]{4+5+2\sqrt{20}}-\sqrt[2]{4+5-2\sqrt{20}}\\ 2+\sqrt{5}-(2-\sqrt{5})\\ 2\sqrt{5}\end{array}$$

1. Tentukan nilai x dari ${}^{x}log{(}^{2}log{(}^{x}log(8+7x)))=0$!

Jawaban

$$\begin{array}{cc}{}^{x}log{(}^{2}log{(}^{x}log(8+7x)))& =0\\ {}^{2}log{(}^{x}log(8+7x))& =x^0\\ {}^{2}log{(}^{x}log(8+7x))& =1\\ {}^{x}log(8+7x)& =2^1\\ {}^{x}log(8+7x)& =2\\ 8+7x& =x^2\\ x^2-7x-8& =0\\ (x-8)(x+1)& =0\\ x=8& \text{ atau }x=-1\\ \text{karena syarat }x>0\text{ maka hanya }x=8\text{ terpenuhi }\end{array}$$

1. Tentukan nilai $(1+\frac{1}{x}{)}^{x+1}$ dari $(1+\frac{1}{8}{)}^8$!

Jawaban

$$\begin{array}{cc}(1+\frac{1}{x}{)}^{x+1}& =(1+\frac{1}{8}{)}^8\\ & =(\frac{9}{8}{)}^8\\ & =(\frac{8}{9}{)}^{-8}\\ & =(1-\frac{1}{9}{)}^{-9+1}\\ & =(1+\frac{1}{-9}{)}^{-9+1}\\ & =-9\end{array}$$

1. Tentukan nilai $2^{x-y}+\frac{1}{2^{x-y}}$ jika 2^x+2^y=6 dan 2^x+y=3!

Jawaban

$$\begin{array}{c}\text{misalkan }{2}^x=a\text{ dan }{2}^y=b\\ 2^x+2^y& =6\\ a+b& =6\\ 2^{x+y}& =3\\ 2^x\cdot 2^y& =3\\ ab& =3\\ 2^{x-y}+\frac{1}{2^{x-y}}& =\frac{2^x}{2^y}+\frac{2^y}{2^x}\\ & =\frac{a}{b}+\frac{b}{a}\\ & =\frac{a^2+b^2}{ab}\\ & =\frac{(a+b{)}^2-2ab}{ab}\\ & =\frac{6^2-2(3)}{3}\\ & =\frac{30}{3}\\ & =10\end{array}$$

1. Tentukan nilai x jika 9^x2-4+2x²=0!

9^x2-4+2x² = 0

9^x2+2x² = 4

(3²)^x2+2x² = 3+1

3^2x2+2x² = 3¹+1

2x²=1

2x²-1=0

($\sqrt{2}$x-1)($\sqrt{2}$x+1)=0

x=$\frac{\sqrt{2}}{2}$ atau x=-$\frac{\sqrt{2}}{2}$

1. Tentukan nilai x jika 64^x2-5+3x²=0!

64^x2-5+3x² = 0

64^x2+3x² = 5

(4³)^x2+3x² = 4+1

4^3x2+3x² = 4¹+1

3x²=1

3x²-1=0

($\sqrt{3}$x-1)($\sqrt{3}$x+1)=0

x=$\frac{\sqrt{3}}{3}$ atau x=-$\frac{\sqrt{3}}{3}$