Soal yang Akan Dibahas
Jika $ \log \frac{a^2}{b^2} = 18 $ , maka $ \log \left( 5\sqrt[3]{\frac{8b}{a}} \right) = ..... $
A). $ -2 \, $ B). $ -1 \, $ C). $ 0 \, $ D). $ 1 \, $ E). $ 2 \, $
A). $ -2 \, $ B). $ -1 \, $ C). $ 0 \, $ D). $ 1 \, $ E). $ 2 \, $
$\spadesuit $ Konsep Dasar
*). Sifat-sifat logaritma :
(i). $ {}^a \log b^n = n. {}^a \log b $
(ii). $ {}^a \log (b.c) = {}^a \log b + {}^a \log c $
*). SIfat-sifat eksponen :
(i). $ \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} $
(ii). $ \sqrt{ab} = \sqrt{a}\sqrt{b} $ dan $ \left(\frac{a}{b} \right)^n = \left( \frac{b}{a} \right)^{-n} $
(iii). $ \sqrt[n]{a} = a^\frac{1}{n} $ dan $ (a^m)^n = a^{m.n} $
*). Sifat-sifat logaritma :
(i). $ {}^a \log b^n = n. {}^a \log b $
(ii). $ {}^a \log (b.c) = {}^a \log b + {}^a \log c $
*). SIfat-sifat eksponen :
(i). $ \sqrt{\frac{a}{b}} = \frac{\sqrt{a}}{\sqrt{b}} $
(ii). $ \sqrt{ab} = \sqrt{a}\sqrt{b} $ dan $ \left(\frac{a}{b} \right)^n = \left( \frac{b}{a} \right)^{-n} $
(iii). $ \sqrt[n]{a} = a^\frac{1}{n} $ dan $ (a^m)^n = a^{m.n} $
$\clubsuit $ Pembahasan
*).Menyederhanakan yang diketahui :
$\begin{align} \log \frac{a^2}{b^2} & = 18 \\ \log \left( \frac{a }{b } \right)^2 & = 18 \\ \log \left( \frac{b}{a } \right)^{-2} & = 18 \\ -2. \log \frac{b}{a } & = 18 \, \, \, \, \, \, \, \text{(bagi -2)} \\ \log \frac{b}{a } & = -9 \end{align} $
*). Menyelesaikan soal :
$ \begin{align} \log \left( 5\sqrt[3]{\frac{8b}{a}} \right) & = \log \left( 5\sqrt[3]{8} \sqrt[3]{\frac{b}{a}} \right) \\ & = \log \left( 5.2 \left( \frac{b}{a } \right)^{\frac{1}{3}} \right) \\ & = \log \left( 10. \left( \frac{b}{a } \right)^{\frac{1}{3}} \right) \\ & = \log 10 + \log \left( \frac{b}{a } \right)^{\frac{1}{3}} \\ & = \log 10 + \frac{1}{3} . \log \left( \frac{b}{a } \right) \\ & = 1 + \frac{1}{3} . (-9) \\ & = 1 - 3 = -2 \end{align} $
Jadi, nilai $ \log \left( 5\sqrt[3]{\frac{8b}{a}} \right) = -2 . \, \heartsuit $
*).Menyederhanakan yang diketahui :
$\begin{align} \log \frac{a^2}{b^2} & = 18 \\ \log \left( \frac{a }{b } \right)^2 & = 18 \\ \log \left( \frac{b}{a } \right)^{-2} & = 18 \\ -2. \log \frac{b}{a } & = 18 \, \, \, \, \, \, \, \text{(bagi -2)} \\ \log \frac{b}{a } & = -9 \end{align} $
*). Menyelesaikan soal :
$ \begin{align} \log \left( 5\sqrt[3]{\frac{8b}{a}} \right) & = \log \left( 5\sqrt[3]{8} \sqrt[3]{\frac{b}{a}} \right) \\ & = \log \left( 5.2 \left( \frac{b}{a } \right)^{\frac{1}{3}} \right) \\ & = \log \left( 10. \left( \frac{b}{a } \right)^{\frac{1}{3}} \right) \\ & = \log 10 + \log \left( \frac{b}{a } \right)^{\frac{1}{3}} \\ & = \log 10 + \frac{1}{3} . \log \left( \frac{b}{a } \right) \\ & = 1 + \frac{1}{3} . (-9) \\ & = 1 - 3 = -2 \end{align} $
Jadi, nilai $ \log \left( 5\sqrt[3]{\frac{8b}{a}} \right) = -2 . \, \heartsuit $
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