Soal yang Akan Dibahas
Jika $ 2^x = a $ dan $ 2^y = b $ dengan $ x , \, y > 0 $ , maka $ \frac{2x+3y}{x+2y} = ... $
A). $\frac{3}{5} \, $ B). $\frac{5}{3} \, $ C). $ 1 + {}^{ab} \log ab^2 \, $ D). $1 + {}^{ab} \log a^2b \, $ E). $1 + {}^{ab^2} \log ab $
A). $\frac{3}{5} \, $ B). $\frac{5}{3} \, $ C). $ 1 + {}^{ab} \log ab^2 \, $ D). $1 + {}^{ab} \log a^2b \, $ E). $1 + {}^{ab^2} \log ab $
$\spadesuit $ Konsep Dasar Logaritma
*). Definisi Logaritma :
$ a^c = b \rightarrow c = {}^a \log b $
*). Sifat-sifat Logaritma :
$ {}^a \log b^n = n. {}^a \log b $
$ {}^a \log (b.c) = {}^a \log b + {}^a \log c $
$ \frac{{}^p \log b }{{}^p \log a} = {}^a \log b $
*). Definisi Logaritma :
$ a^c = b \rightarrow c = {}^a \log b $
*). Sifat-sifat Logaritma :
$ {}^a \log b^n = n. {}^a \log b $
$ {}^a \log (b.c) = {}^a \log b + {}^a \log c $
$ \frac{{}^p \log b }{{}^p \log a} = {}^a \log b $
$\clubsuit $ Pembahasan
*). Menyederhanakan soal :
$\begin{align} 2^x & = a \rightarrow x = {}^2 \log a \\ 2^y & = b \rightarrow y = {}^2 \log b \end{align} $
*). Menyelesaikan soal :
$ \begin{align} \frac{2x+3y}{x+2y} & = \frac{2.{}^2 \log a +3.{}^2 \log b}{{}^2 \log a+2.{}^2 \log b} \\ & = \frac{{}^2 \log a^2 + {}^2 \log b^3}{{}^2 \log a+{}^2 \log b^2} \\ & = \frac{{}^2 \log a^2b^3 }{{}^2 \log a b^2} \\ & = {}^{a b^2} \log a^2b^3 \\ & = {}^{a b^2} \log [(ab^2).(ab)] \\ & = {}^{a b^2} \log (ab^2) + {}^{a b^2} \log (ab) \\ & = 1 + {}^{a b^2} \log ab \end{align} $ .
Jadi, nilai $ \frac{2x+3y}{x+2y} = 1 + {}^{a b^2} \log ab . \, \heartsuit $
*). Menyederhanakan soal :
$\begin{align} 2^x & = a \rightarrow x = {}^2 \log a \\ 2^y & = b \rightarrow y = {}^2 \log b \end{align} $
*). Menyelesaikan soal :
$ \begin{align} \frac{2x+3y}{x+2y} & = \frac{2.{}^2 \log a +3.{}^2 \log b}{{}^2 \log a+2.{}^2 \log b} \\ & = \frac{{}^2 \log a^2 + {}^2 \log b^3}{{}^2 \log a+{}^2 \log b^2} \\ & = \frac{{}^2 \log a^2b^3 }{{}^2 \log a b^2} \\ & = {}^{a b^2} \log a^2b^3 \\ & = {}^{a b^2} \log [(ab^2).(ab)] \\ & = {}^{a b^2} \log (ab^2) + {}^{a b^2} \log (ab) \\ & = 1 + {}^{a b^2} \log ab \end{align} $ .
Jadi, nilai $ \frac{2x+3y}{x+2y} = 1 + {}^{a b^2} \log ab . \, \heartsuit $
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