Logaritma

A. Pendahuluan

Silahkan kunjungi alamat ini di sini

B. Sifat-Sifat

$\color{blue}\begin{array}{|l|l|}\hline \qquad\qquad\color{black}\textrm{Logaritma}\\\hline \color{black}^{a}\log b=c\: \Rightarrow \: a^{c}=b\\\hline \bullet \quad \color{black}^{a}\log x+\: ^{a}\log y=\: ^{a}\log xy\\\hline \bullet \quad \color{black}^{a}\log x-\: ^{a}\log y=\: ^{a}\log \displaystyle \frac{x}{y}\\\hline \bullet \quad ^{a}\log x=\: \displaystyle \frac{^{m}\log x}{^{m}\log a}\\\hline \bullet \quad ^{a}\log b\: \times \: ^{b}\log c=\: ^{a}\log c\\\hline \bullet \quad ^{a^{m}}\log b^{n}=\displaystyle \frac{n}{m}\times \: ^{a}\log b\\\hline \bullet \quad \displaystyle a^{\: {^{a}}\log b}=b\\\hline \bullet \quad ^{a}\log b=\displaystyle \frac{1}{^{b}\log a}\\\hline \bullet \quad ^{a}\log 1=0\\\hline \bullet \quad \color{black}^a\log a=1\\\hline \begin{cases} a\neq 0 &\\ a>0&(\textrm{bilangan pokok}) \\ x,y>0 & (\textrm{numerus}) \end{cases}\\\hline \end{array}$.

$\LARGE\colorbox{yellow}{CONTOH SOAL}$.

$\begin{array}{ll}\\ 1.&\textrm{Hitunglah}\\ &\textrm{a}.\quad ^{36}\log 6\\ &\textrm{b}.\quad ^{8}\log \displaystyle \frac{1}{4}\\ &\textrm{c}.\quad ^{2}\log 3+\: ^{2}\log 12-\: ^{2}\log 9\\ &\textrm{d}.\quad ^{16}\log \sqrt[3]{25}\times \, ^{5}\log \displaystyle \frac{1}{4}\\\\ &\textbf{Jawab}:\\ &\begin{aligned}\textrm{a}.\quad &^{36}\log 6=\: ^{6^{2}}\log 6^{1}=\displaystyle \frac{1}{2}\times \, ^{6}\log 6\\ &=\color{red}\displaystyle \frac{1}{2}\times \: \color{black}\underset{\color{red}1}{\underbrace{^{6}\log 6}}=\color{red}\displaystyle \frac{1}{2}\times 1\color{black}=\color{red}\displaystyle \frac{1}{2}\\ &\textrm{atau}\\ &^{36}\log 6=\displaystyle \frac{1}{^{6}\log 36}=\displaystyle \frac{1}{^{6^{.^{1}}}\log 6^{2}}\\ &=\color{red}\displaystyle \frac{1}{\frac{2}{1}\times \, \color{black}^{6}\log 6}\color{black}=\color{red}\displaystyle \frac{1}{2\times 1}\color{black}=\color{red}\displaystyle \frac{1}{2} \\ \textrm{b}.\quad &^{8}\log 4=\: ^{2^{3}}\log 2^{2}=\displaystyle \frac{2}{3}\times \color{black}\underset{\color{red}1}{\underbrace{^{2}\log 2}}\\ &=\color{red}\displaystyle \frac{2}{3}\times 1\color{black}=\color{red}\displaystyle \frac{2}{3}\\ \end{aligned} \\ &\begin{aligned} \textrm{c}.\quad &^{2}\log 3+\: ^{2}\log 12-\: ^{2}\log 9\\ &=\: ^{2}\log \color{blue}\left ( \displaystyle \frac{3\times 12}{9} \right )\color{black}=\: ^{2}\log \color{blue}\displaystyle \frac{36}{9}\\ &=\: ^{2}\log \color{blue}4\color{black}=\: ^{2}\log \color{blue}2^{2}\color{black}=\color{blue}2\color{black}\times \, \color{black}\underset{\color{red}1}{\underbrace{^{2}\log 2}}\\ &=\color{red}2\times 1\color{black}=\color{red}2\\ \end{aligned}\\ &\begin{aligned}\textrm{d}.\quad &^{16}\log \sqrt[3]{25}\times \, ^{5}\log \displaystyle \frac{1}{4}\\ &=\: ^{4^{.^{2}}}\log 5^{.^{\frac{2}{3}}}\times \: ^{.5^{.^{1}}}\log 4^{-1}\\ &=\left (\displaystyle \frac{\left ( \displaystyle \frac{2}{3} \right )}{2}\times \, ^{4}\log 5 \right )\times \left (\displaystyle \frac{-1}{1}\times \: ^{5}\log 4 \right )\\ &=\: \color{red}\displaystyle \frac{1}{3}\times -1\color{black}\times \, ^{4}\log 5\times \, ^{5}\log 4\\ &=\color{red}-\displaystyle \frac{1}{3}\times \, \color{black}^{4}\log 4,\qquad \color{blue}(\textrm{ingat})\\ &=\color{red}-\displaystyle \frac{1}{3}\times \color{black}\underset{\color{red}1}{\underbrace{^{4}\log 4}}\\ &=\color{red}-\displaystyle \frac{1}{3}\times 1\\ &=\color{red}-\displaystyle \frac{1}{3} \end{aligned} \end{array}$.

C. Persamaan Logaritma

Bentuk-bentuk persamaan logaritma secara umum adalah persamaan dengan numerus ataupun bilangan basis/pokok yang memuat variabel x.

$\begin{aligned}1.\quad &^{a}\log f(x)=\: ^{a}\log p\\ 2.\quad&^{a}\log f(x)=\: ^{b}\log f(x)\\ 3.\quad &^{a}\log f(x)=\: ^{a}\log g(x)\\ 4.\quad &^{h(x)}\log f(x)=\: ^{h(x)}\log g(x)\\ 5.\quad&A\left (^{a}\log f(x) \right )^{2}+B\: \left (^{a}\log f(x) \right )+C=0\\ \end{aligned}$.

$\LARGE\colorbox{yellow}{CONTOH SOAL}$.

$\begin{array}{ll}\\ 2.&\textrm{Hitunglah}\\ &\textrm{a}.\quad \log (x-5)=\log 3\\ &\textrm{b}.\quad \log (2x^{2}-x)=1\\ &\textrm{c}.\quad ^{3}\log (x^{2}-3x+5)=1\\\\ &\textrm{Jawab}:\\ &\textrm{Yang dibahas hanya no.2a, yaitu}:\\ &\begin{aligned}\textrm{a}.\quad &\textrm{Diketahui numerus}:x-5\\ &\textrm{1. Syarat numerus}:\: f(x)>0\\ &\quad x-5>0\Leftrightarrow x>5\\ &\textrm{2. Persamaan}\\ &\quad \log (x-5)=\log 3\\ &\quad \Leftrightarrow x-5=3\\ &\quad \Leftrightarrow x=8\\ &\textrm{3. Simpulan}\\ &\quad \textrm{Karena}\: \: x>5,\\ &\quad \textrm{maka}\: \: x=8\: \: \textrm{memenuhi}\\ &\quad \textrm{Jadi},\: \: \textrm{HP}=\left \{ 8 \right \} \end{aligned} \end{array}$.