Soal yang Akan Dibahas
Diberikan $ y > x > 0 $. Jika $ {}^9 \log (y^2 - x^2) = a $ dan $ {}^{x+y} \log 3 = b $ ,
maka $ {}^{27} \log (y-x) = ... $
A). $ \frac{3ab + 1}{2a} \, $ B). $ \frac{3ab - 1}{2b} \, $ C). $ \frac{2ab - 1}{3b} \, $
D). $ \frac{2ab + 1}{3a} \, $ E). $ \frac{2ab - 1}{3a} $
A). $ \frac{3ab + 1}{2a} \, $ B). $ \frac{3ab - 1}{2b} \, $ C). $ \frac{2ab - 1}{3b} \, $
D). $ \frac{2ab + 1}{3a} \, $ E). $ \frac{2ab - 1}{3a} $
$\spadesuit $ Konsep Dasar :
*). Sifat-sifat Logaritma :
$ {}^a log (b.c) = {}^a \log b + {}^a \log c $
$ {}^a \log b = y \rightarrow {}^b \log a = \frac{1}{y} $
$ {}^{a^m} \log b = \frac{1}{m} . {}^a \log b $
*). Pemfaktoran : $ a^2 - b^2 = (a-b)(a+b) $
*). Sifat-sifat Logaritma :
$ {}^a log (b.c) = {}^a \log b + {}^a \log c $
$ {}^a \log b = y \rightarrow {}^b \log a = \frac{1}{y} $
$ {}^{a^m} \log b = \frac{1}{m} . {}^a \log b $
*). Pemfaktoran : $ a^2 - b^2 = (a-b)(a+b) $
$\clubsuit $ Pembahasan
*). Mengubah bentuk $ {}^{x+y} \log 3 = b $ :
$\begin{align} {}^{x+y} \log 3 & = b \rightarrow {}^3 \log (x + y) = \frac{1}{b} \end{align} $
*). Menentukan nilai $ {}^3 \log (y - x) $ :
$\begin{align} {}^9 \log (y^2 - x^2) & = a \\ {}^{3^2} \log (y^2 - x^2) & = a \\ \frac{1}{2} . {}^3 \log (y^2 - x^2) & = a \\ {}^3 \log (y^2 - x^2) & = 2a \\ {}^3 \log (y-x)(y+x) & = 2a \\ {}^3 \log (y-x) + {}^3 \log (y+x) & = 2a \\ {}^3 \log (y-x) + \frac{1}{b} & = 2a \\ {}^3 \log (y-x) & = 2a - \frac{1}{b} \\ {}^3 \log (y-x) & = \frac{2ab}{b} - \frac{1}{b} \\ {}^3 \log (y-x) & = \frac{2ab - 1}{b} \end{align} $
*). Menentukan nilai $ {}^{27} \log (y-x) $ :
$\begin{align} {}^{27} \log (y-x) & = {}^{3^3 } \log (y-x) \\ & = \frac{1}{3} . {}^{3} \log (y-x) \\ & = \frac{1}{3} . \frac{2ab - 1}{b} \\ & = \frac{2ab - 1}{3b} \end{align} $
Jadi, nilai $ {}^{27} \log (y-x) = \frac{2ab - 1}{3b} . \, \heartsuit $
*). Mengubah bentuk $ {}^{x+y} \log 3 = b $ :
$\begin{align} {}^{x+y} \log 3 & = b \rightarrow {}^3 \log (x + y) = \frac{1}{b} \end{align} $
*). Menentukan nilai $ {}^3 \log (y - x) $ :
$\begin{align} {}^9 \log (y^2 - x^2) & = a \\ {}^{3^2} \log (y^2 - x^2) & = a \\ \frac{1}{2} . {}^3 \log (y^2 - x^2) & = a \\ {}^3 \log (y^2 - x^2) & = 2a \\ {}^3 \log (y-x)(y+x) & = 2a \\ {}^3 \log (y-x) + {}^3 \log (y+x) & = 2a \\ {}^3 \log (y-x) + \frac{1}{b} & = 2a \\ {}^3 \log (y-x) & = 2a - \frac{1}{b} \\ {}^3 \log (y-x) & = \frac{2ab}{b} - \frac{1}{b} \\ {}^3 \log (y-x) & = \frac{2ab - 1}{b} \end{align} $
*). Menentukan nilai $ {}^{27} \log (y-x) $ :
$\begin{align} {}^{27} \log (y-x) & = {}^{3^3 } \log (y-x) \\ & = \frac{1}{3} . {}^{3} \log (y-x) \\ & = \frac{1}{3} . \frac{2ab - 1}{b} \\ & = \frac{2ab - 1}{3b} \end{align} $
Jadi, nilai $ {}^{27} \log (y-x) = \frac{2ab - 1}{3b} . \, \heartsuit $
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