Question
Manakah yang merupakan bentuk eksponen dari \(^3\log_{}{243}=5\)?
- \(3^4 = 243\)
- \(3^5 = 243\)
- \(5^3 = 243\)
- \(243^3 = 5\)
- \(243^3 = 15\)
Answer: B
\(\color{blue} \text{Gunakan sifat } ^a\log_{}{b} = c \Leftrightarrow a^c=b\)
\(^3\log_{}{243}=5 \Leftrightarrow 3^5 = 243\)
Question
Manakah yang merupakan bentuk logaritma dari \(2^7 = 128\)?
- \(^2\log_{}{128} = 7\)
- \(^7\log_{}{128} = 2\)
- \(^2\log_{}{7} = 128\)
- \(^7\log_{}{2} = 128\)
- \(^7\log_{}{2} = 130\)
Answer: A
\(\color{blue}\text{Gunakan sifat } a^c = b \Leftrightarrow ^a \log_{}{b} = c\)
\(2^7 = 128 \Leftrightarrow ^2\log_{}{128}=7\)
Question
Hasil perhitungan dari \(^4\log_{}{8} + ^4\log_{}{16} \:-\: ^4\log_{}{2}\) adalah …
Answer: C
\(\color{blue}\text{Gunakan sifat: }\)
\(\color{blue}^a\log_{}{b} + ^a\log_{}{c} = ^a\log_{}{bc}\)
\(\color{blue}^a\log_{}{b} \:-\: ^a\log_{}{c} = ^a\log_{}{\frac{b}{c}}\)
\(^4\log_{}{8} + ^4\log_{}{16} \:-\: ^4\log_{}{2}\)
\(^4\log_{}{\frac{8\times \cancelto{8}{16}}{\cancel{2}}}\)
\(^4\log_{}{64}\)
\(^4\log_{}{4^3}\)
\(3 \cdot ^4\log_{}{4}=3\)
Question
Hasil dari \(\dfrac{(^3\log_{}{49})^2 – (^3\log_{}{7})^2}{(^3\log_{}{7})^2}\) adalah …
Answer: C
\(\color{blue} a^2 \:-\: b^2 = (a + b)(a \:-\: b)\)
\(\dfrac{(^3\log_{}{7^2} + ^3\log_{}{7})(^3\log_{}{49} \:-\: ^3\log_{}{7})}{(^3\log_{}{7})^2}\)
\(\dfrac{^3\log{}{7^3}\cdot ^3\log_{}{\frac{49}{7}}}{^3\log{}{7}\cdot ^3\log{}{7}}\)
\(\dfrac{ 3 \cdot ^3\log{}{7}\cdot ^3\log_{}{7}}{^3\log{}{7}\cdot ^3\log_{}{7}}\)
\(3\)
Question
Jika \(^4\log_{}{5} = p \text{ dan } ^4\log_{}{28}=q\) maka nilai \(^4\log_{}{70}\) adalah …
Answer: A
\(\text{Sederhanakan : }\)
\(^4\log_{}{5} = p\)
\(^{2^2}\log_{}{5^1} = p\)
\(\dfrac{1}{2} \cdot ^2\log_{}{5} = p\)
\(^2\log_{}{5} = 2p\)
\(^4\log_{}{28}=q\)
\(^{2^2}\log_{}{(2^2 \times 7)}=q\)
\(^{2^2}\log_{}{2^2} + ^{2^2}\log_{}{7^1} = q\)
\(\frac{2}{2}\cdot ^{2}\log_{}{2} + \frac{1}{2}\cdot ^{2}\log_{}{7} = q\)
\(1+ \frac{1}{2}\cdot ^{2}\log_{}{7} = q\)
\(\frac{1}{2}\cdot ^{2}\log_{}{7} = q \:-\: 1\)
\(^{2}\log_{}{7} = 2q-2\)
\(\text{Selanjutnya:}\)
\(^4\log_{}{70} = \dfrac{^m\log_{}{70}}{^m\log_{}{4}}\)
\(\dfrac{^m\log_{}{(2\times 5\times 7)}}{^m\log_{}{2^2}}\)
\(\dfrac{^m\log_{}{2}+ ^m\log_{}{5}+ ^m\log_{}{7}}{2\cdot ^m\log_{}{2}}\)
\(\text{Pilih basis m yang cocok, m = 2}\)
\(\dfrac{^2\log_{}{2}+ ^2\log_{}{5}+ ^2\log_{}{7}}{2\cdot ^2\log_{}{2}}\)
\(\dfrac{1+ 2p + 2q \:-\: 2}{2}\)
\(p+ q \:-\: \frac{1}{2}\)
Question
Jika \(^4\log_{}{6} = m + 1\) maka \(^9\log_{}{8}\) = …
Answer: B
\(\text{Sederhanakan dahulu }^4\log_{}{6} = m + 1\)
\(^{2^2}\log_{}{(2\times 3)} = m + 1\)
\(^{2^2}\log_{}{2} + ^{2^2}\log_{}{3}= m + 1\)
\(\frac{1}{2}\cdot ^{2}\log_{}{2} + \frac{1}{2}\cdot ^{2}\log_{}{3}= m + 1\)
\(\frac{1}{2} + \frac{1}{2}\cdot ^{2}\log_{}{3}= m + 1\)
\(1+ ^{2}\log_{}{3}= 2m + 2\)
\(^{2}\log_{}{3}= 2m + 1\)
\(\text{Selanjutnya } ^9\log_{}{8} = ^{3^2}\log_{}{2^3}\)
\(\frac{3}{2}\cdot ^3\log_{}{2}\)
\(\frac{3}{2}\cdot \dfrac{1}{^2\log_{}{3}}\)
\(\dfrac{3}{2 \times ^2\log_{}{3}}\)
\(\dfrac{3}{2(2m+1)}\)
\(\dfrac{3}{4m+2}\)
Question
Nilai dari \(8^{^2\log_{}{5}} \times 4^{\frac{1}{^5\log_{}{2}}} = \dotso\)
Answer: A
\({(2^3)}^{^2\log_{}{5}} \times {(2^2)}^{^2\log_{}{5}}\)
\(2^{3\cdot\: ^2\log_{}{5}}\times 2^{2\cdot\: ^2\log_{}{5}}\)
\(2^{^2\log_{}{5^3}} \times 2^{^2\log_{}{5^2}}\)
\(2^{^2\log_{}{125}} \times 2^{^2\log_{}{25}}\)
\(125 \times 25\)
\(3125\)
Question
Hasil dari \(^2\log_{}{^2\log_{}{\sqrt{\sqrt{2}}}}\) adalah …
Answer: B
\(^2\log_{}{^2\log_{}{\sqrt{\sqrt{2}}}}\)
\(^2\log_{}\:^2\log_{} 2^{\frac{1}{4}}\)
\(^2\log_{}\frac{1}{4}\)
\(^2\log_{} 2^{-2}\)
\(-2\)
Question
Hasil dari \(\dfrac{1}{^3\log_{}{2}} + \dfrac{1}{^9\log_{}{4}} \:-\: \dfrac{4}{^9\log_{}{16}}\) adalah …
Answer: A
\(\dfrac{1}{^3\log_{}{2}} + \dfrac{1}{^9\log_{}{4}} \:-\: \dfrac{4}{^9\log_{}{16}}\)
\(\:^2\log 3 + ^4\log 9 \:-\: 4\cdot ^{16}\log 9\)
\(\: ^2\log 3 +^{2^2}\log 3^2 \:-\: 4\cdot ^{2^4}\log 3^2\)
\(\: ^2\log 3 + \frac{2}{2}\cdot ^2\log 3 \:-\: 4\cdot \frac{2}{4}\cdot ^2\log 3\)
\(\: ^2\log 3 + ^2\log 3 \:-\: 2\cdot ^2\log 3\)
\(0\)
Question
Jika \(^2\log_{}{3} = p\) dan \(^5\log_{}{4} = q\) maka \(^5\log_{}{6}=\dotso\)
Answer: A
\(^2\log_{}{3} = p\)
\(^5\log_{}{4} = q\)
\(^5\log_{}{2^2} = q\)
\(2\cdot ^5\log_{}{2} = q\)
\(^5\log_{}{2} = \dfrac{q}{2}\)
\(^5\log_{}{6} = \dfrac{^2\log (2 \times 3)}{^2\log 5}\:\:\:\:\:\color{blue}\text{(pilih basis 2)}\)
\(\dfrac{^2\log 2 + ^2\log 3}{^2\log 5}\)
\(\dfrac{1 + p}{\frac{2}{q}}\)
\(\dfrac{q + pq}{2}\)
\(\frac{1}{2}(pq + q)\)