Soal yang Akan Dibahas
Jika $ f(x) = \frac{{}^4 \log x}{1 - 2.{}^4 \log x} $ , maka $ f(2a) + f\left( \frac{2}{a} \right) = .... $
A). $ - a \, $ B). $ -1 \, $ C). $ 0 \, $ D). $ 1 \, $ E). $ a $
A). $ - a \, $ B). $ -1 \, $ C). $ 0 \, $ D). $ 1 \, $ E). $ a $
$\spadesuit $ Konsep Dasar Logaritma
*). Sifat-sifat Logartima :
$ {}^a \log (bc) = {}^a \log b + {}^a \log c $
$ {}^a \log \frac{b}{c} = {}^a \log b - {}^a \log c $
$ {{}^a}^m \log b^n = \frac{n}{m} . {}^a \log b $
Sehingga $ {}^4 \log 2 = {{}^2}^2 \log 2 = \frac{1}{2} {}^2 \log 2 = \frac{1}{2} $
*). Sifat-sifat Logartima :
$ {}^a \log (bc) = {}^a \log b + {}^a \log c $
$ {}^a \log \frac{b}{c} = {}^a \log b - {}^a \log c $
$ {{}^a}^m \log b^n = \frac{n}{m} . {}^a \log b $
Sehingga $ {}^4 \log 2 = {{}^2}^2 \log 2 = \frac{1}{2} {}^2 \log 2 = \frac{1}{2} $
$\clubsuit $ Pembahasan
*). Menentukan nilai $ f(2a) $ dengan $ f(x) = \frac{{}^4 \log x}{1 - 2.{}^4 \log x} $
$\begin{align} f(2a) & = \frac{{}^4 \log 2a}{1 - 2.{}^4 \log 2a} \\ & = \frac{{}^4 \log 2 + {}^4 \log a}{1 - 2.({}^4 \log 2 + {}^4 \log a)} \\ & = \frac{\frac{1}{2} + {}^4 \log a}{1 - 2.(\frac{1}{2} + {}^4 \log a)} \\ & = \frac{\frac{1}{2} + {}^4 \log a}{1 - 1 - 2 {}^4 \log a } \\ & = \frac{\frac{1}{2} + {}^4 \log a}{ - 2 {}^4 \log a } \\ & = \frac{-\frac{1}{2} - {}^4 \log a}{ 2 {}^4 \log a } \end{align} $
*). Menentukan nilai $ f(\frac{2}{a}) $ dengan $ f(x) = \frac{{}^4 \log x}{1 - 2.{}^4 \log x} $
$\begin{align} f(2a) & = \frac{{}^4 \log \frac{2}{a}}{1 - 2.{}^4 \log \frac{2}{a}} \\ & = \frac{{}^4 \log 2 - {}^4 \log a}{1 - 2.({}^4 \log 2 - {}^4 \log a)} \\ & = \frac{\frac{1}{2} - {}^4 \log a}{1 - 2.(\frac{1}{2} - {}^4 \log a)} \\ & = \frac{\frac{1}{2} - {}^4 \log a}{1 - 1 + 2 {}^4 \log a } \\ & = \frac{\frac{1}{2} - {}^4 \log a}{ 2 {}^4 \log a } \end{align} $
*). Menentukan nilai $ f(2a) + f(\frac{2}{a}) $ :
$\begin{align} f(2a) + f\left( \frac{2}{a} \right) & = \frac{-\frac{1}{2} - {}^4 \log a}{ 2 {}^4 \log a } + \frac{\frac{1}{2} - {}^4 \log a}{ 2 {}^4 \log a } \\ & = \frac{ - 2{}^4 \log a}{ 2 {}^4 \log a } \\ & = -1 \end{align} $
Jadi, nilai $ f(2a) + f\left( \frac{2}{a} \right) = -1 . \, \heartsuit $
*). Menentukan nilai $ f(2a) $ dengan $ f(x) = \frac{{}^4 \log x}{1 - 2.{}^4 \log x} $
$\begin{align} f(2a) & = \frac{{}^4 \log 2a}{1 - 2.{}^4 \log 2a} \\ & = \frac{{}^4 \log 2 + {}^4 \log a}{1 - 2.({}^4 \log 2 + {}^4 \log a)} \\ & = \frac{\frac{1}{2} + {}^4 \log a}{1 - 2.(\frac{1}{2} + {}^4 \log a)} \\ & = \frac{\frac{1}{2} + {}^4 \log a}{1 - 1 - 2 {}^4 \log a } \\ & = \frac{\frac{1}{2} + {}^4 \log a}{ - 2 {}^4 \log a } \\ & = \frac{-\frac{1}{2} - {}^4 \log a}{ 2 {}^4 \log a } \end{align} $
*). Menentukan nilai $ f(\frac{2}{a}) $ dengan $ f(x) = \frac{{}^4 \log x}{1 - 2.{}^4 \log x} $
$\begin{align} f(2a) & = \frac{{}^4 \log \frac{2}{a}}{1 - 2.{}^4 \log \frac{2}{a}} \\ & = \frac{{}^4 \log 2 - {}^4 \log a}{1 - 2.({}^4 \log 2 - {}^4 \log a)} \\ & = \frac{\frac{1}{2} - {}^4 \log a}{1 - 2.(\frac{1}{2} - {}^4 \log a)} \\ & = \frac{\frac{1}{2} - {}^4 \log a}{1 - 1 + 2 {}^4 \log a } \\ & = \frac{\frac{1}{2} - {}^4 \log a}{ 2 {}^4 \log a } \end{align} $
*). Menentukan nilai $ f(2a) + f(\frac{2}{a}) $ :
$\begin{align} f(2a) + f\left( \frac{2}{a} \right) & = \frac{-\frac{1}{2} - {}^4 \log a}{ 2 {}^4 \log a } + \frac{\frac{1}{2} - {}^4 \log a}{ 2 {}^4 \log a } \\ & = \frac{ - 2{}^4 \log a}{ 2 {}^4 \log a } \\ & = -1 \end{align} $
Jadi, nilai $ f(2a) + f\left( \frac{2}{a} \right) = -1 . \, \heartsuit $
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