Soal 1
Berikut ini adalah sifat-sifat logaritma:
(1) \(^\text{a}\log_{}{\text{b}} + ^\text{a}\log_{}{\text{c}} = ^\text{a}\log_{}{\text{bc}} \)
(2) \(^\text{a}\log_{}{\text{b}}\cdot ^\text{b}\log_{}{\text{c}} \cdot ^\text{c}\log_{}{\text{d}} = ^\text{a}\log_{}{\text{d}} \)
(3) \(^{\text{a}^{\text{n}}}\log_{}{\text{b}^{\text{m}}} = \frac{n}{m}\cdot ^\text{a}\log_{}{\text{b}} \)
(4) \(^{\text{a}^{\text{n}}}\log_{}{\text{b}} = ^\text{a}\log_{}{\text{b}^{\frac{1}{\text{n}}}} \)
(5) \(\dfrac{1}{^\text{a}\log_{}{\text{b}}} = ^\text{b}\log_{}{\text{a}}\)
(6) \(\text{a}^{^\text{a}\log_{}{\text{b}}} = \text{a}^{\text{b}}\)
Sifat logaritma yang salah adalah nomor …
(A) 1 dan 2
(B) 2 dan 3
(C) 2 dan 4
(D) 3 dan 5
(E) 3 dan 6
Jawaban: E
Koreksi nomor 3:
\(^{\text{a}^{\text{n}}}\log_{}{\text{b}^{\text{m}}} = \frac{m}{n}\cdot ^\text{a}\log_{}{\text{b}} \)
Koreksi nomor 6:
\(\text{a}^{^\text{a}\log_{}{\text{b}}} = \text{b}\)
Soal 2
\(\dfrac{4\cdot \log_{}{\dfrac{1}{25}} + 4\cdot \log_{}{3} \:-\:\log_{}{\dfrac{1}{81}}}{{\log_{}{\dfrac{1}{(3\cdot \sqrt [3] {3})}}} + \dfrac{4}{3}\cdot\log_{}{5}} =\dotso\)
(A) −6
(B) −4
(C) −2
(D) 2
(E) 4
Jawaban: A
\(\dfrac{4\cdot \log_{}{5^{-2}} + \log_{}{3^4}\:-\:\log_{}{3^{-4}}}{\log_{}{3^{-\frac{4}{3}} + \log_{}{5^{\frac{4}{3}}}}}\)
\(\dfrac{\log_{}{\dfrac{5^{-8}\cdot 3^4}{3^{-4}}}}{\log_{}{3^{-\frac{4}{3}} \cdot 5^{\frac{4}{3}}}}\)
\(\dfrac{\log_{}{5^{-8}\cdot 3^8}}{\log_{}{3^{-\frac{4}{3}} \cdot 5^{\frac{4}{3}}}}\)
\(\dfrac{\log_{}{(\frac{3}{5})^8}}{\log_{}{(\frac{3}{5})^{-\frac{4}{3}}}}\)
\(\dfrac{8}{-\frac{4}{3}}\)
\(-6\)
Soal 3
\(\left(\dfrac{1}{^{\frac{1}{8}}\log_{}{9}} + \dfrac{1}{\log_{}{9}}+\dfrac{1}{2\cdot ^4\log_{}{3}}\right)\times ^5\log_{}{9} = \dotso\)
(A) \(\dfrac{1}{3}\)
(B) \(\dfrac{1}{4}\)
(C) \(1\)
(D) \(2\)
(E) \(5\)
Jawaban: C
\(\left(\dfrac{1}{^{2^{-3}}\log_{}{9}} + \dfrac{1}{\log_{}{9}}+\dfrac{1}{^4\log_{}{3^2}}\right)\times ^5\log_{}{9}\)
\(\left(^9\log_{}{2^{-3}} + ^9\log_{}{10} + ^9\log_{}{4}\right)\times ^5\log_{}{9}\)
\(\left(^9\log_{}{(2^{-3}\times 10 \times 4)}\right)\times ^5\log_{}{9}\)
\(\left(^9\log_{}{\dfrac{40}{8}}\right)\times ^5\log_{}{9}\)
\(^9\log_{}{5}\times ^5\log_{}{9}\)
\(^9\log_{}{9} = 1\)
Soal 4
Jika \(\log_{}{80} = a\), maka \(^{40}\log_{}{800} = \dotso\)
(A) \(\dfrac{a + 3}{2a \:-\: 3}\)
(B) \(\dfrac{a + 1}{2a +1}\)
(C) \(\dfrac{2a + 3}{2a +1}\)
(D) \(\dfrac{3a + 3}{2a \:-\: 1}\)
(E) \(\dfrac{3a + 3}{2a +1}\)
Jawaban: E
\(^{40}\log_{}{800} = \dfrac{\log_{}{800}}{\log_{}{40}}\)
\(^{40}\log_{}{800} = \dfrac{\log_{}{(80 \times 10)}}{\log_{}{\frac{80}{2}}}\)
\(^{40}\log_{}{800} = \dfrac{\log_{}{80} + \log_{}{10}}{\log_{}{80}\:-\:\log_{}{2}}\)
\(^{40}\log_{}{800} = \dfrac{a + 1}{a\:-\:\log_{}{2}}\)
Diketahui bahwa \(\log_{}{80} = a\)
\(\log_{}{(8\times 10)} = a\)
\(\log_{}{8} + \log_{}{10}= a\)
\(\log_{}{2^3} + 1 = a\)
\(3\cdot \log_{}{2} = a\:-\:1\)
\(\log_{}{2} = \dfrac{a}{3}\:-\:\dfrac{1}{3}\)
\(^{40}\log_{}{800} = \dfrac{a + 1}{a\:-\:(\dfrac{a}{3}\:-\:\dfrac{1}{3})}\)
\(^{40}\log_{}{800} = \dfrac{a + 1}{\dfrac{2a}{3}+\dfrac{1}{3}}\)
\(^{40}\log_{}{800} = \dfrac{3a + 3}{2a +1}\)
Soal 5
Jika \(^{2r}\log_{}{24} = ^{3r}\log_{}{36}\), maka nilai dari \(5r\) adalah …
(A) 40
(B) 50
(C) 60
(D) 70
(E) 80
Jawaban: C
\(\text{Misal }^{2r}\log_{}{24} = a\)
\(24 = (2r)^a\)
\(24 = 2^a\cdot r^a\)
\(r^a = \dfrac{24}{2^a}\)
\(^{3r}\log_{}{36} = a\)
\(36 = (3r)^a\)
\(36 = 3^a\cdot r^a\)
\(36 = 3^a \cdot \dfrac{24}{2^a}\)
\(\dfrac{36}{24} = \left(\dfrac{3}{2}\right)^a\)
\(\dfrac{3}{2} = \left(\dfrac{3}{2}\right)^a\)
\(a = 1\)
\(r^a = \dfrac{24}{2^a}\)
\(r^1 = \dfrac{24}{2^1} \rightarrow r = 12\)
Nilai dari \(5r = 5 \times 12 = 60\)
Soal 6
Jika diketahui \(\begin{cases}\log_{}{x^4\cdot \sqrt [3] {y^2}} = 25\\\\\log_{}{\dfrac{y^3}{x\cdot \sqrt [3] {x}}} = 24\end{cases}\), maka nilai \(\log_{}{x^{16}y^{-7}} = \dotso\)
(A) \(3\)
(B) \(4\)
(C) \(5\)
(D) \(6\)
(E) \(7\)
Jawaban: A
\(\begin{cases}\log_{}{x^4\cdot \sqrt [3] {y^2}} = 25\\\\\log_{}{\dfrac{y^3}{x\cdot \sqrt [3] {x}}} = 24\end{cases}\),
Kurangkan kedua persamaan di atas, sehingga diperoleh:
\(\log_{}{x^4\cdot \sqrt [3] {y^2}}\:-\:\log_{}{\dfrac{y^3}{x\cdot \sqrt [3] {x}}} = 25\:-\:24\)
\(\log_{}{x^4\cdot y^{\frac{2}{3}}}\:-\:\log_{}{\dfrac{y^3}{x^{\frac{4}{3}}}} = 1\)
\(\log_{}{x^4\cdot y^{\frac{2}{3}}}\:-\:\log_{}{y^3\cdot x^{-\frac{4}{3}}} = 1\)
\(\log_{}{\dfrac{x^4\cdot y^{\frac{2}{3}}}{y^3\cdot x^{-\frac{4}{3}}}}= 1\)
\(\log_{}{x^{\frac{16}{3}}\cdot y^{-\frac{7}{3}}} = 1\)
\(\log_{}{(x^{16}\cdot y^{-7})^{\frac{1}{3}}} = 1\)
\(\frac{1}{3}\log_{}{x^{16}\cdot y^{-7}} = 1\)
\(\log_{}{x^{16}\cdot y^{-7}} = 3\)
Soal 7
Diketahui \(^2\log_{}{3} = a, ^2\log_{}{5} = b, \text{ dan } ^3\log_{}{7} = c\). Nilai \(^{28}\log_{}{160}\), jika dinyatakan dalam \(a, b, \text{ dan } c\) adalah …
(A) \(\dfrac{5 + 2b}{1 + ac}\)
(B) \(\dfrac{3+b}{2 + ac}\)
(C) \(\dfrac{5 +2b}{2 + ac}\)
(D) \(\dfrac{5 +b}{2 + ac}\)
(E) \(\dfrac{5 \:-\:b}{2 + ac}\)
Jawaban: D
\(^{28}\log_{}{160} = \dfrac{^3\log_{}{160}}{^3\log_{}{28}}\)
\(^{28}\log_{}{160} = \dfrac{^3\log_{}{2^5\cdot 5}}{^3\log_{}{2^2\cdot 7}}\)
\(^{28}\log_{}{160} = \dfrac{^3\log_{}{2^5} +^3\log_{}{5} }{^3\log_{}{2^2} +^3\log_{}{7} }\)
\(^{28}\log_{}{160} = \dfrac{5\cdot ^3\log_{}{2} +^3\log_{}{5} }{2\cdot ^3\log_{}{2} +^3\log_{}{7} }\)
\(^{28}\log_{}{160} = \dfrac{5\cdot \dfrac{1}{a} +^3\log_{}{5} }{2\cdot \dfrac{1}{a} + c}\)
Cara mencari nilai \(^3\log_{}{5}\)
\(^3\log_{}{5} = \dfrac{^2\log_{}{5}}{^2\log_{}{3}} = \dfrac{b}{a}\)
\(^{28}\log_{}{160} = \dfrac{5\cdot \dfrac{1}{a} +\dfrac{b}{a}}{2\cdot \dfrac{1}{a} + c}\)
\(\color{blue}\text{kalikan pembilang dan penyebut dengan }a\)
\(^{28}\log_{}{160} = \dfrac{5 +b}{2 + ac}\)
Soal 8
\(\left(\sqrt{2}\right)^{^{0,5}\log_{}{\frac{1}{81}}} = \dotso\)
(A) 3
(B) 4
(C) 5
(D) 6
(E) 9
Jawaban: E
\(\left(\sqrt{2}\right)^{^{\frac{1}{2}}\log_{}{\frac{1}{3^4}}}\)
\(\left(2^{\frac{1}{2}}\right)^{^{2^{-1}}\log_{}{3^{-4}}}\)
\(2^{\frac{1}{2}\cdot 4\cdot ^2\log_{}{3}}\)
\(2^{2\cdot ^2\log_{}{3}}\)
\(2^{^2\log_{}{3^2}}\)
\(\color{blue} a^{^a\log_{}{b}} = b\)
\(3^2 = 9\)
Soal 9
\(^2\log_{}{64} + 1000^{\log_{}{2}}\:-\:^5\log_{}{25\sqrt{5}} + ^{\frac{1}{4}}\log_{}{8} + \log_{}{1} = \dotso\)
(A) 8
(B) 9
(C) 10
(D) 11
(E) 12
Jawaban: C
\(^2\log_{}{2^6} + \left(10^3\right)^{\log_{}{2}}\:-\:^5\log_{}{5^{\frac{5}{2}}} + ^{\frac{1}{2^2}}\log_{}{2^3} + 0 \)
\(6 + \left(10\right)^{3\cdot \log_{}{2}}\:-\:\dfrac{5}{2} + ^{2^{-2}}\log_{}{2^3}\)
\(6 + 10^{\log_{}{2^3}}\:-\:\dfrac{5}{2} \:-\:\dfrac{3}{2}\)
\(6 + 2^3\:-\:\dfrac{5}{2} \:-\:\dfrac{3}{2}\)
\(6 + 8\:-\:\dfrac{8}{2}\)
\(10\)
Soal 10
Jika \(\log_{}{\left(\dfrac{a}{b}\right)^{\frac{3}{2}}} = 27\) maka nilai \(\log_{}{\sqrt[3]{\dfrac{b^2}{a^2}}}= \dotso\)
(A) −12
(B) −10
(C) 2
(D) 4
(E) 12
Jawaban: A
\(\log_{}{\left(\dfrac{a}{b}\right)^{\frac{3}{2}}} = 27\)
\(\dfrac{3}{2}\cdot \log_{}{\dfrac{a}{b}} = 27\)
\(\log_{}{\dfrac{a}{b}} = \dfrac{2}{\cancel{3}}\times \cancelto {9}{27}\)
\(\log_{}{\dfrac{a}{b}} = 18\)
\(\log_{}{\sqrt[3]{\dfrac{b^2}{a^2}}}\)
\(\log_{}{\left(\dfrac{b}{a}\right)^{\frac{2}{3}}}\)
\(\log_{}{\left(\dfrac{a}{b}\right)^{-\frac{2}{3}}}\)
\(-\dfrac{2}{3}\cdot \log_{}{\left(\dfrac{a}{b}\right)}\)
\(-\dfrac{2}{\cancel{3}}\cdot \cancelto{6}{18}\)
\(-12\)
