,
Logaritma merupakan invers dari perpangkatan
$^{a}\textrm{log}\: b=x\Leftrightarrow a^{x}=b$
dengan:
* a bilangan pokok (numerus), dengan $a> 0$ dan $a\neq 0$
* b numerus, yaitu bilangan yang dicari logaritmanya, dengan $b> 0$
* x hasil logaritma (merupakan eksponen dari a yang menghasilkan b)
* $^{a}\textrm{log}\: b$ dibaca logaritma b dengan bilangan pokok a
* Jika a = 10, bilangan pokok biasanya tidak ditulis, contoh $^{10}\textrm{log}\: 5$ ditulis $log\: 5$
Rumus-rumus Logaritma
# ${log}\: a\: \cdot \: b={log}\: a+{log}\: b$
# ${log}\: \frac{a}{b}={log}\: a-{log}\: b$
# ${log}\: a^{n}=n\: \cdot \: {log}\: a$
# $a^{^{a}\textrm{log}\: b}=b$
# $^{a}\textrm{log}\: b=\frac{log\: b}{log\: a}=\frac{1}{^{b}\textrm{log}\: a}$
# $^{a^{m}}\textrm{log}\; b^{n}=^{a}\textrm{log}\: b^{\frac{n}{m}}$
# $^{a}\textrm{log}\: b\: \cdot \: ^{b}\textrm{log}\: c=\: ^{a}\textrm{log}\: c$
Soal dan Pembahasan
1. UN 2010 P12
Hasil dari $\frac{^{3}\textrm{log}\: 5\: \cdot \: ^{\sqrt{5}}\textrm{log}\: 9+\: ^{8}\textrm{log}\: 2}{^{2}\textrm{log}\: 12-\: ^{2}\textrm{log}\: 3}$ = ....
(A) $\frac{4}{6}$
(B) $\frac{7}{6}$
(C) $\frac{5}{3}$
(D) $\frac{13}{6}$
(E) $\frac{26}{6}$
Pembahasan:
$\frac{^{3}\textrm{log}\: 5\: \cdot \: ^{\sqrt{5}}\textrm{log}\: 9+\: ^{8}\textrm{log}\: 2}{^{2}\textrm{log}\: 12-\: ^{2}\textrm{log}\: 3}$
$=\frac{^{3}\textrm{log}\: 5\: \cdot \: ^{5^{\frac{1}{2}}}\textrm{log}\: 3^{2}+\: ^{2^{3}}\textrm{log}\: 2}{^{2}\textrm{log}\: \frac{12}{3}}$
$=\frac{^{3}\textrm{log}\: 5\: \cdot \: ^{5}\textrm{log}\: 3^{\frac{2}{\frac{1}{2}}}+\: ^{2}\textrm{log}\: 2^{\frac{1}{3}}}{^{2}\textrm{log}\: 4}$
$=\frac{^{{\color{DarkRed} 3}}\textrm{log}\: {\color{Red} 5}\: \cdot \: ^{{\color{Red} 5}}\textrm{log}\: {\color{DarkRed} 3}^4+\: ^{{\color{Blue} 2}}\textrm{log}\: {\color{Blue} 2}^{\frac{1}{3}}}{^{{\color{Pink} 2}}\textrm{log}\: {\color{Pink} 2}^{2}}$
$=\frac{4+\frac{1}{3}}{2}$
$=\frac{\frac{12}{3}+\frac{1}{3}}{2}$
$=\frac{\frac{13}{3}}{2}$
$=\frac{13}{6}$
Jawaban _____________________________ (D)
2. UMPTN 2001 Matematika Dasar Rayon B
Jika $^{10}\textrm{log}\: x=b$ maka $^{10x}\textrm{log}\: 100$ = ....
(A) $\frac{1}{b+1}$
(B) $\frac{2}{b+1}$
(C) $\frac{1}{b}$
(D) $\frac{2}{b}$
(E) $\frac{2}{10b}$
Pembahasan:
$^{10}\textrm{log}\: x=b\Leftrightarrow 10^{b}=x$
maka
$^{10x}\textrm{log}\: 100$
$=\: ^{10\left (10^{b} \right )}\textrm{log}\: 100$
$=\: ^{100^{b}}\textrm{log}\: 100$
$=\: ^{100}\textrm{log}\: 100^{\frac{1}{b}}$
$=\frac{1}{b}$
Jawaban _____________________________ (C)
3. UMPTN 2001 Matematika Dasar Rayon B
Jika $a=0,111....$, maka nilai $^{a}\textrm{log}\: 729$= ....
(A) -5
(B) -4
(C) -3
(D) 4
(E) 5
Pembahasan:
$a=0,111......\: \: \: \: \: \: \: (x10)$
$10a=1,111......$ _
$-9a=-1$
$a=\frac{1}{9}$
maka $^{a}\textrm{log}\: 729$
$=\: ^{\frac{1}{9}}\textrm{log}\: 729$
$=\: ^{9^{-1}}\textrm{log}\: 9^{3}$
$=\: ^{9}\textrm{log}\: 9^{-3}$
$=-3$
Jawaban _____________________________ (C)
4. SPMB 2003 Regional I
Jika $\frac{1}{2}\: ^{2}\textrm{log}\: x+\: ^{4}\textrm{log}\: \sqrt{y}=\: ^{4}\textrm{log}\: z^{2}$, maka $z^{2}$= ....
(A) $x\sqrt{y}$
(B) $x^{2}\sqrt{y}$
(C) $xy$
(D) $x\sqrt[4]{y}$
(E) $x^{2}\sqrt[4]{y}$
Pembahasan:
$\frac{1}{2}\: ^{2}\textrm{log}\: x+\: ^{4}\textrm{log}\: \sqrt{y}=\: ^{4}\textrm{log}\: z^{2}$
$^{2}\textrm{log}\: x^{\frac{1}{2}}+\: ^{4}\textrm{log}\: \sqrt{y}=\: ^{4}\textrm{log}\: z^{2}$
$^{2^{2}}\textrm{log}\: x+\: ^{4}\textrm{log}\: \sqrt{y}=\: ^{4}\textrm{log}\: z^{2}$
$^{4}\textrm{log}\: x+\: ^{4}\textrm{log}\: \sqrt{y}=\: ^{4}\textrm{log}\: z^{2}$
$^{4}\textrm{log}\: x\sqrt{y}=\: ^{4}\textrm{log}\: z^{2}$
$x\sqrt{y}=z^{2}$
Jawaban _____________________________ (A)
$^{a}\textrm{log}\: b=x\Leftrightarrow a^{x}=b$
dengan:
* a bilangan pokok (numerus), dengan $a> 0$ dan $a\neq 0$
* b numerus, yaitu bilangan yang dicari logaritmanya, dengan $b> 0$
* x hasil logaritma (merupakan eksponen dari a yang menghasilkan b)
* $^{a}\textrm{log}\: b$ dibaca logaritma b dengan bilangan pokok a
* Jika a = 10, bilangan pokok biasanya tidak ditulis, contoh $^{10}\textrm{log}\: 5$ ditulis $log\: 5$
Rumus-rumus Logaritma
# ${log}\: a\: \cdot \: b={log}\: a+{log}\: b$
# ${log}\: \frac{a}{b}={log}\: a-{log}\: b$
# ${log}\: a^{n}=n\: \cdot \: {log}\: a$
# $a^{^{a}\textrm{log}\: b}=b$
# $^{a}\textrm{log}\: b=\frac{log\: b}{log\: a}=\frac{1}{^{b}\textrm{log}\: a}$
# $^{a^{m}}\textrm{log}\; b^{n}=^{a}\textrm{log}\: b^{\frac{n}{m}}$
# $^{a}\textrm{log}\: b\: \cdot \: ^{b}\textrm{log}\: c=\: ^{a}\textrm{log}\: c$
Soal dan Pembahasan
1. UN 2010 P12
Hasil dari $\frac{^{3}\textrm{log}\: 5\: \cdot \: ^{\sqrt{5}}\textrm{log}\: 9+\: ^{8}\textrm{log}\: 2}{^{2}\textrm{log}\: 12-\: ^{2}\textrm{log}\: 3}$ = ....
(A) $\frac{4}{6}$
(B) $\frac{7}{6}$
(C) $\frac{5}{3}$
(D) $\frac{13}{6}$
(E) $\frac{26}{6}$
Pembahasan:
$\frac{^{3}\textrm{log}\: 5\: \cdot \: ^{\sqrt{5}}\textrm{log}\: 9+\: ^{8}\textrm{log}\: 2}{^{2}\textrm{log}\: 12-\: ^{2}\textrm{log}\: 3}$
$=\frac{^{3}\textrm{log}\: 5\: \cdot \: ^{5^{\frac{1}{2}}}\textrm{log}\: 3^{2}+\: ^{2^{3}}\textrm{log}\: 2}{^{2}\textrm{log}\: \frac{12}{3}}$
$=\frac{^{3}\textrm{log}\: 5\: \cdot \: ^{5}\textrm{log}\: 3^{\frac{2}{\frac{1}{2}}}+\: ^{2}\textrm{log}\: 2^{\frac{1}{3}}}{^{2}\textrm{log}\: 4}$
$=\frac{^{{\color{DarkRed} 3}}\textrm{log}\: {\color{Red} 5}\: \cdot \: ^{{\color{Red} 5}}\textrm{log}\: {\color{DarkRed} 3}^4+\: ^{{\color{Blue} 2}}\textrm{log}\: {\color{Blue} 2}^{\frac{1}{3}}}{^{{\color{Pink} 2}}\textrm{log}\: {\color{Pink} 2}^{2}}$
$=\frac{4+\frac{1}{3}}{2}$
$=\frac{\frac{12}{3}+\frac{1}{3}}{2}$
$=\frac{\frac{13}{3}}{2}$
$=\frac{13}{6}$
Jawaban _____________________________ (D)
2. UMPTN 2001 Matematika Dasar Rayon B
Jika $^{10}\textrm{log}\: x=b$ maka $^{10x}\textrm{log}\: 100$ = ....
(A) $\frac{1}{b+1}$
(B) $\frac{2}{b+1}$
(C) $\frac{1}{b}$
(D) $\frac{2}{b}$
(E) $\frac{2}{10b}$
Pembahasan:
$^{10}\textrm{log}\: x=b\Leftrightarrow 10^{b}=x$
maka
$^{10x}\textrm{log}\: 100$
$=\: ^{10\left (10^{b} \right )}\textrm{log}\: 100$
$=\: ^{100^{b}}\textrm{log}\: 100$
$=\: ^{100}\textrm{log}\: 100^{\frac{1}{b}}$
$=\frac{1}{b}$
Jawaban _____________________________ (C)
3. UMPTN 2001 Matematika Dasar Rayon B
Jika $a=0,111....$, maka nilai $^{a}\textrm{log}\: 729$= ....
(A) -5
(B) -4
(C) -3
(D) 4
(E) 5
Pembahasan:
$a=0,111......\: \: \: \: \: \: \: (x10)$
$10a=1,111......$ _
$-9a=-1$
$a=\frac{1}{9}$
maka $^{a}\textrm{log}\: 729$
$=\: ^{\frac{1}{9}}\textrm{log}\: 729$
$=\: ^{9^{-1}}\textrm{log}\: 9^{3}$
$=\: ^{9}\textrm{log}\: 9^{-3}$
$=-3$
Jawaban _____________________________ (C)
4. SPMB 2003 Regional I
Jika $\frac{1}{2}\: ^{2}\textrm{log}\: x+\: ^{4}\textrm{log}\: \sqrt{y}=\: ^{4}\textrm{log}\: z^{2}$, maka $z^{2}$= ....
(A) $x\sqrt{y}$
(B) $x^{2}\sqrt{y}$
(C) $xy$
(D) $x\sqrt[4]{y}$
(E) $x^{2}\sqrt[4]{y}$
Pembahasan:
$\frac{1}{2}\: ^{2}\textrm{log}\: x+\: ^{4}\textrm{log}\: \sqrt{y}=\: ^{4}\textrm{log}\: z^{2}$
$^{2}\textrm{log}\: x^{\frac{1}{2}}+\: ^{4}\textrm{log}\: \sqrt{y}=\: ^{4}\textrm{log}\: z^{2}$
$^{2^{2}}\textrm{log}\: x+\: ^{4}\textrm{log}\: \sqrt{y}=\: ^{4}\textrm{log}\: z^{2}$
$^{4}\textrm{log}\: x+\: ^{4}\textrm{log}\: \sqrt{y}=\: ^{4}\textrm{log}\: z^{2}$
$^{4}\textrm{log}\: x\sqrt{y}=\: ^{4}\textrm{log}\: z^{2}$
$x\sqrt{y}=z^{2}$
Jawaban _____________________________ (A)
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