Pembahasan Logaritma UTUL UGM 2017 Matematika Dasar Kode 723

Soal yang Akan Dibahas
Jika $ {}^2 \log (a-b) = 4 $ , maka $ {}^4 \log \left( \frac{2}{\sqrt{a}+\sqrt{b}} + \frac{2}{\sqrt{a}-\sqrt{b}} \right) = .... $
A). $\frac{{}^2 \log a \, - 4 }{4} \, $ B). $\frac{{}^2 \log a \, + 4 }{4} \, $
C). $\frac{{}^2 \log a \, - 2 }{2} \, $ D). $\frac{{}^2 \log a \, + 2 }{2} \, $
E). $\frac{{}^2 \log a \, - 1 }{2} $

$\spadesuit $ Konsep Dasar Logaritma :
*). Definisi Logaritma
$ {}^x \log y = z \rightarrow y = x^z $
*). Sifat-sifat logaritma :
i). $ {{}^a}^m \log b = \frac{1}{m} {}^a \log b $
ii). $ {}^a \log \frac{b}{c} = {}^a \log b - {}^a \log c $
iii). $ {}^a \log b^n = n. {}^a \log b $

$\clubsuit $ Pembahasan
*). Menentukan nilai $ a - b $ dari definisi log :
$\begin{align} {}^2 \log (a-b) & = 4 \rightarrow a - b = 2^4 = 16 \end{align} $
*). Menyelesaikan soal dengan sifat-sifat :
$ \begin{align} & {}^4 \log \left( \frac{2}{\sqrt{a}+\sqrt{b}} + \frac{2}{\sqrt{a}-\sqrt{b}} \right) \\ & = {{}^2}^2 \log \left( \frac{2(\sqrt{a}-\sqrt{b})+2(\sqrt{a}+\sqrt{b})}{(\sqrt{a}+\sqrt{b})(\sqrt{a}-\sqrt{b})}\right) \\ & = \frac{1}{2} . {}^2 \log \left( \frac{4\sqrt{a}}{a-b}\right) \\ & = \frac{1}{2} . {}^2 \log \left( \frac{4\sqrt{a}}{16}\right) \\ & = \frac{1}{2} . {}^2 \log \left( \frac{a^\frac{1}{2}}{4}\right) \\ & = \frac{1}{2} . \left( {}^2 \log a^\frac{1}{2} - {}^2 \log 4 \right) \\ & = \frac{1}{2} . \left( \frac{1}{2} {}^2 \log a - 2 \right) \\ & = \frac{1}{2} . \left[ \frac{1}{2} ( {}^2 \log a - 4) \right] \\ & = \frac{1}{4} . ( {}^2 \log a - 4) \\ & = \frac{{}^2 \log a - 4}{4} \end{align} $ .
Jadi, $ {}^4 \log \left( \frac{2}{\sqrt{a}+\sqrt{b}} + \frac{2}{\sqrt{a}-\sqrt{b}} \right) = \frac{{}^2 \log a - 4}{4} . \, \heartsuit $

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