1. UM UGM 2017 Kode 723
Jika $\large ^{2}\textrm{log}\: \left ( a-b \right )=4$, maka $\large ^{4}\textrm{log}\: \left ( \frac{2}{\sqrt{a}+\sqrt{b}}+\frac{2}{\sqrt{a}-\sqrt{b}} \right )$ = ....
(A) $\large \frac{^{2}\textrm{log}\: a-4}{4}$
(B) $\large \frac{^{2}\textrm{log}\: a+4}{4}$
(C) $\large \frac{^{2}\textrm{log}\: a-2}{4}$
(D) $\large \frac{^{2}\textrm{log}\: a+2}{4}$
(E) $\large \frac{^{2}\textrm{log}\: a-1}{4}$
Pembahasan:
$\large ^{2}\textrm{log}\: \left ( a-b \right )=4$
$\large \left ( a-b \right )=2^{4}$
$\large =16$
Pergunakan akar sekawan
$\large \frac{a}{b+c\sqrt{d}}\: \cdot \: \frac{b-c\sqrt{d}}{b-c\sqrt{d}}$
$\large \frac{a}{b-c\sqrt{d}}\: \cdot \: \frac{b+c\sqrt{d}}{b+c\sqrt{d}}$
$\large ^{4}\textrm{log}\: \left ( \frac{2}{\sqrt{a}+\sqrt{b}}+\frac{2}{\sqrt{a}-\sqrt{b}} \right )$
$\large =\: \large ^{4}\textrm{log}\: \left ( \frac{2}{\sqrt{a}+\sqrt{b}}\: \cdot \: \frac{\sqrt{a}-\sqrt{b}}{\sqrt{a}-\sqrt{b}}+\frac{2}{\sqrt{a}-\sqrt{b}}\: \cdot \: \frac{\sqrt{a}+\sqrt{b}}{\sqrt{a}+\sqrt{b}} \right )$
$\large =\: \large ^{4}\textrm{log}\: \left ( \frac{2\sqrt{a}-2\sqrt{b}}{a-b}+\frac{2\sqrt{a}+2\sqrt{b}}{a-b} \right )$
$\large =\: \large ^{4}\textrm{log}\: \left ( \frac{4\sqrt{a}}{a-b}\right )$
$\large =\: \large ^{4}\textrm{log}\: \left ( \frac{4\sqrt{a}}{16}\right )$
$\large =\: \large ^{4}\textrm{log}\: \left ( \frac{\sqrt{a}}{4}\right )$
$\large =\: \large ^{4}\textrm{log}\: \sqrt{a}-\: \large ^{4}\textrm{log} \: 4$
$\large =\: \large ^{2^{2}}\textrm{log}\: a^{\frac{1}{2}}-1$
$\large =\: \large ^{2}\textrm{log}\: a^{\frac{1}{4}}-1$
$\large =\frac{1}{4}\: \large ^{2}\textrm{log}\: a-1$
$\large =\frac{1}{4}\: \large ^{2}\textrm{log}\: a-\frac{4}{4}$
$\large =\frac{\: \large ^{2}\textrm{log}\: a-4}{4}$
Jawaban ________________________________ (A)
2. UM UGM 2017 Kode 823
Jika $\large \frac{3-3\sqrt{2}}{\sqrt{3}-\sqrt{6}}=b$, maka $\large ^{b}\textrm{log}\: 9$ = ....
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
Pembahasan:
Pergunakan perkalian sekawan
$\large \frac{a}{b+\sqrt{c}}=\frac{a}{b+\sqrt{c}}\: \cdot \: \frac{b-\sqrt{c}}{b-\sqrt{c}}$
$\large \frac{a}{b-\sqrt{c}}=\frac{a}{b-\sqrt{c}}\: \cdot \: \frac{b+\sqrt{c}}{b+\sqrt{c}}$
$\large b=\frac{3-3\sqrt{2}}{\sqrt{3}-\sqrt{6}}$
$\large b=\frac{3-3\sqrt{2}}{\sqrt{3}-\sqrt{6}}\: \cdot \: \frac{\sqrt{3}+\sqrt{6}}{\sqrt{3}+\sqrt{6}}$
$\large b=\frac{3\left (1-\sqrt{2} \right )\left ( \sqrt{3} +\sqrt{6}\right ) }{3-{6}}$
$\large b=\frac{3\left (1-\sqrt{2} \right )\left ( \sqrt{3} +\sqrt{6}\right ) }{-3}$
$\large b=-\left ( 1-\sqrt{2} \right )\left ( \sqrt{3}+\sqrt{6} \right )$
$\large b=-\left ( \sqrt{3}+\sqrt{6}-\sqrt{6}-\sqrt{12} \right )$
$\large b=-\left ( \sqrt{3}-\sqrt{4}\cdot \sqrt{3} \right )$
$\large b=-\left ( \sqrt{3}-2\sqrt{3} \right )$
$\large b=-\left ( -\sqrt{3} \right )$
$\large b=\sqrt{3}$
maka
$\large ^{b}\textrm{log}\: 9$
$\large =^{\sqrt{3}}\textrm{log}\: 3^{2}$
$\large =\: ^{3^{\frac{1}{2}}}\textrm{log}\: 3^{2}$
$\large =\: ^{3}\textrm{log}\: 3^{4}$
$\large =4$
Jawaban ________________________________ (D)
3. UM UGM 2017 Kode 823
Jika $\large u=2^{x}$ dan $\large ^{u}\textrm{log}\: \left ( 2^{2x}-2^{x-2} \right )=3$, maka $\large 3^{x}$ = ....
(A) 3
(B) 1
(C) $\large \frac{1}{3}$
(D) $\large \frac{1}{9}$
(E) $\large \frac{1}{27}$
Pembahasan:
$\large ^{u}\textrm{log}\: 2^{2x}-\: ^{u}\textrm{log}\: 2^{x-2}=3$
$\large ^{2^{x}}\textrm{log}\: 2^{2x}-\: ^{2^{x}}\textrm{log}\: 2^{x-2}=3$
$\large ^{2}\textrm{log}\: 2^{\frac{2x}{x}}-\: ^{2}\textrm{log}\: 2^{\frac{x-2}{x}}=3$
$\large \frac{2x}{x}-\frac{x-2}{x}=3$
$\large \frac{x+2}{x}=3$
$\large x+2=3x$
$\large 2x=2$
$\large x=1$
maka
$\large 3^{x}$
$\large =3^{1}$
$\large =3$
Jawaban ________________________________ (A)
Jika $\large ^{2}\textrm{log}\: \left ( a-b \right )=4$, maka $\large ^{4}\textrm{log}\: \left ( \frac{2}{\sqrt{a}+\sqrt{b}}+\frac{2}{\sqrt{a}-\sqrt{b}} \right )$ = ....
(A) $\large \frac{^{2}\textrm{log}\: a-4}{4}$
(B) $\large \frac{^{2}\textrm{log}\: a+4}{4}$
(C) $\large \frac{^{2}\textrm{log}\: a-2}{4}$
(D) $\large \frac{^{2}\textrm{log}\: a+2}{4}$
(E) $\large \frac{^{2}\textrm{log}\: a-1}{4}$
Pembahasan:
$\large ^{2}\textrm{log}\: \left ( a-b \right )=4$
$\large \left ( a-b \right )=2^{4}$
$\large =16$
Pergunakan akar sekawan
$\large \frac{a}{b+c\sqrt{d}}\: \cdot \: \frac{b-c\sqrt{d}}{b-c\sqrt{d}}$
$\large \frac{a}{b-c\sqrt{d}}\: \cdot \: \frac{b+c\sqrt{d}}{b+c\sqrt{d}}$
$\large ^{4}\textrm{log}\: \left ( \frac{2}{\sqrt{a}+\sqrt{b}}+\frac{2}{\sqrt{a}-\sqrt{b}} \right )$
$\large =\: \large ^{4}\textrm{log}\: \left ( \frac{2}{\sqrt{a}+\sqrt{b}}\: \cdot \: \frac{\sqrt{a}-\sqrt{b}}{\sqrt{a}-\sqrt{b}}+\frac{2}{\sqrt{a}-\sqrt{b}}\: \cdot \: \frac{\sqrt{a}+\sqrt{b}}{\sqrt{a}+\sqrt{b}} \right )$
$\large =\: \large ^{4}\textrm{log}\: \left ( \frac{2\sqrt{a}-2\sqrt{b}}{a-b}+\frac{2\sqrt{a}+2\sqrt{b}}{a-b} \right )$
$\large =\: \large ^{4}\textrm{log}\: \left ( \frac{4\sqrt{a}}{a-b}\right )$
$\large =\: \large ^{4}\textrm{log}\: \left ( \frac{4\sqrt{a}}{16}\right )$
$\large =\: \large ^{4}\textrm{log}\: \left ( \frac{\sqrt{a}}{4}\right )$
$\large =\: \large ^{4}\textrm{log}\: \sqrt{a}-\: \large ^{4}\textrm{log} \: 4$
$\large =\: \large ^{2^{2}}\textrm{log}\: a^{\frac{1}{2}}-1$
$\large =\: \large ^{2}\textrm{log}\: a^{\frac{1}{4}}-1$
$\large =\frac{1}{4}\: \large ^{2}\textrm{log}\: a-1$
$\large =\frac{1}{4}\: \large ^{2}\textrm{log}\: a-\frac{4}{4}$
$\large =\frac{\: \large ^{2}\textrm{log}\: a-4}{4}$
Jawaban ________________________________ (A)
2. UM UGM 2017 Kode 823
Jika $\large \frac{3-3\sqrt{2}}{\sqrt{3}-\sqrt{6}}=b$, maka $\large ^{b}\textrm{log}\: 9$ = ....
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
Pembahasan:
Pergunakan perkalian sekawan
$\large \frac{a}{b+\sqrt{c}}=\frac{a}{b+\sqrt{c}}\: \cdot \: \frac{b-\sqrt{c}}{b-\sqrt{c}}$
$\large \frac{a}{b-\sqrt{c}}=\frac{a}{b-\sqrt{c}}\: \cdot \: \frac{b+\sqrt{c}}{b+\sqrt{c}}$
$\large b=\frac{3-3\sqrt{2}}{\sqrt{3}-\sqrt{6}}$
$\large b=\frac{3-3\sqrt{2}}{\sqrt{3}-\sqrt{6}}\: \cdot \: \frac{\sqrt{3}+\sqrt{6}}{\sqrt{3}+\sqrt{6}}$
$\large b=\frac{3\left (1-\sqrt{2} \right )\left ( \sqrt{3} +\sqrt{6}\right ) }{3-{6}}$
$\large b=\frac{3\left (1-\sqrt{2} \right )\left ( \sqrt{3} +\sqrt{6}\right ) }{-3}$
$\large b=-\left ( 1-\sqrt{2} \right )\left ( \sqrt{3}+\sqrt{6} \right )$
$\large b=-\left ( \sqrt{3}+\sqrt{6}-\sqrt{6}-\sqrt{12} \right )$
$\large b=-\left ( \sqrt{3}-\sqrt{4}\cdot \sqrt{3} \right )$
$\large b=-\left ( \sqrt{3}-2\sqrt{3} \right )$
$\large b=-\left ( -\sqrt{3} \right )$
$\large b=\sqrt{3}$
maka
$\large ^{b}\textrm{log}\: 9$
$\large =^{\sqrt{3}}\textrm{log}\: 3^{2}$
$\large =\: ^{3^{\frac{1}{2}}}\textrm{log}\: 3^{2}$
$\large =\: ^{3}\textrm{log}\: 3^{4}$
$\large =4$
Jawaban ________________________________ (D)
3. UM UGM 2017 Kode 823
Jika $\large u=2^{x}$ dan $\large ^{u}\textrm{log}\: \left ( 2^{2x}-2^{x-2} \right )=3$, maka $\large 3^{x}$ = ....
(A) 3
(B) 1
(C) $\large \frac{1}{3}$
(D) $\large \frac{1}{9}$
(E) $\large \frac{1}{27}$
Pembahasan:
$\large ^{u}\textrm{log}\: 2^{2x}-\: ^{u}\textrm{log}\: 2^{x-2}=3$
$\large ^{2^{x}}\textrm{log}\: 2^{2x}-\: ^{2^{x}}\textrm{log}\: 2^{x-2}=3$
$\large ^{2}\textrm{log}\: 2^{\frac{2x}{x}}-\: ^{2}\textrm{log}\: 2^{\frac{x-2}{x}}=3$
$\large \frac{2x}{x}-\frac{x-2}{x}=3$
$\large \frac{x+2}{x}=3$
$\large x+2=3x$
$\large 2x=2$
$\large x=1$
maka
$\large 3^{x}$
$\large =3^{1}$
$\large =3$
Jawaban ________________________________ (A)
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