Logaritma

A. Pendahuluan

Silahkan kunjungi alamat ini di sini

B. Sifat-Sifat

$\begin{array}{|l|}\hline \text{Logaritma}\\ {}^a\log b=c\Rightarrow a^c=b\\ \bullet {}^a\log x+{}^a\log y={}^a\log xy\\ \bullet {}^a\log x-{}^a\log y={}^a\log {\displaystyle \frac{x}{y}}\\ \bullet {}^a\log x={\displaystyle \frac{{}^m\log x}{{}^m\log a}}\\ \bullet {}^a\log b\times {}^b\log c={}^a\log c\\ \bullet {}^{a^m}\log b^n={\displaystyle \frac{n}{m}\times {}^a\log b}\\ \bullet {\displaystyle a^{{}^a\log b}=b}\\ \bullet {}^a\log b={\displaystyle \frac{1}{{}^b\log a}}\\ \bullet {}^a\log 1=0\\ \bullet {}^a\log a=1\\ \{\begin{array}{ll}a\ne 0& \\ a>0& (\text{bilangan pokok})\\ x,y>0& (\text{numerus})\end{array}\\ \hline\end{array}$.

$\text{CONTOH SOAL}$.

$\begin{array}{l}\\ 1.& \text{Hitunglah}\\ & \text{a}.{}^{36}\log 6\\ & \text{b}.{}^8\log {\displaystyle \frac{1}{4}}\\ & \text{c}.{}^2\log 3+{}^2\log 12-{}^2\log 9\\ & \text{d}.{}^{16}\log \sqrt[3]{25}\times {}^5\log {\displaystyle \frac{1}{4}}\\ \\ & \mathbf{\text{Jawab}}:\\ & \begin{array}{rl}\text{a}.& {}^{36}\log 6={}^{6^2}\log 6^1={\displaystyle \frac{1}{2}\times {}^6\log 6}\\ & ={\displaystyle \frac{1}{2}\times \underset{1}{\underbrace{{}^6\log 6}}={\displaystyle \frac{1}{2}\times 1={\displaystyle \frac{1}{2}}}}\\ & \text{atau}\\ & {}^{36}\log 6={\displaystyle \frac{1}{{}^6\log 36}={\displaystyle \frac{1}{{}^{6^{{.}^1}}\log 6^2}}}\\ & ={\displaystyle \frac{1}{\frac{2}{1}\times {}^6\log 6}={\displaystyle \frac{1}{2\times 1}={\displaystyle \frac{1}{2}}}}\\ \text{b}.& {}^8\log 4={}^{2^3}\log 2^2={\displaystyle \frac{2}{3}\times \underset{1}{\underbrace{{}^2\log 2}}}\\ & ={\displaystyle \frac{2}{3}\times 1={\displaystyle \frac{2}{3}}}\end{array}\\ & \begin{array}{rl}\text{c}.& {}^2\log 3+{}^2\log 12-{}^2\log 9\\ & ={}^2\log \left({\displaystyle \frac{3\times 12}{9}}\right)={}^2\log {\displaystyle \frac{36}{9}}\\ & ={}^2\log 4={}^2\log 2^2=2\times \underset{1}{\underbrace{{}^2\log 2}}\\ & =2\times 1=2\end{array}\\ & \begin{array}{rl}\text{d}.& {}^{16}\log \sqrt[3]{25}\times {}^5\log {\displaystyle \frac{1}{4}}\\ & ={}^{4^{{.}^2}}\log 5^{{.}^{\frac{2}{3}}}\times {}^{{.5}^{{.}^1}}\log 4^{-1}\\ & =\left({\displaystyle \frac{\left({\displaystyle \frac{2}{3}}\right)}{2}\times {}^4\log 5}\right)\times \left({\displaystyle \frac{-1}{1}\times {}^5\log 4}\right)\\ & ={\displaystyle \frac{1}{3}\times -1\times {}^4\log 5\times {}^5\log 4}\\ & =-{\displaystyle \frac{1}{3}\times {}^4\log 4,(\text{ingat})}\\ & =-{\displaystyle \frac{1}{3}\times \underset{1}{\underbrace{{}^4\log 4}}}\\ & =-{\displaystyle \frac{1}{3}\times 1}\\ & =-{\displaystyle \frac{1}{3}}\end{array}\end{array}$.

C. Persamaan Logaritma

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

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

$\text{CONTOH SOAL}$.

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