Logarithms
\(^a\log b = c \rightarrow a^c = b\)
\(a > 0, \: b > 0\) dan \(a \neq 1\)
Example 01
Find the value of \(x\) from \(^2 \log x = 3\)
\begin{equation*} \begin{split} & ^2 \log x = 3 \\\\ & x = 2^3 \\\\ & \bbox[5px, border: 2px solid magenta] {x = 8} \end{split} \end{equation*}
Laws of Logarithms
Exponent
\(^a \log c^p = p \:.\: ^a \log c\)
\(^{a^q} \log c = \dfrac{1}{q} \:.\: ^a \log c\)
\(^{a^q} \log c^p = \dfrac{p}{q} \:.\: ^a \log c\)
Sum and Substraction
\(^c\log a + \: ^c\log b = \: ^c \log a \:.\: b\)
\(^c\log a - \: ^c\log b = \: ^c \log \dfrac{a}{b}\)
Multiplication
\(^a\log b \:.\: ^b\log c = \: ^a \log c\)
Reciprocal
\(^a\log b = \dfrac{1}{^b \log a}\)
Exponent Form
\((c)^{^{c}\log a} = a\)
Example 02
\begin{equation*} \begin{split} & ^7\log \frac{1}{49} \\\\ & ^7\log 7^{-2} \\\\ & -2 \:.\: ^7\log 7 \\\\ & \bbox[5px, border: 2px solid magenta] {-2} \end{split} \end{equation*}
Example 03
\begin{equation*} \begin{split} & ^8\log 2 \\\\ & ^{\large{2^3}}\log 2 \\\\ & \frac{1}{3}\:.\: ^2 \log 2 \\\\ & \bbox[5px, border: 2px solid magenta] {\frac{1}{3}} \end{split} \end{equation*}
Example 04
\begin{equation*} \begin{split} & ^{25}\log \frac{1}{625} \\\\ & ^{\large{5^2}}\log 5^{-4} \\\\ & \frac{-4}{2} \: .\: ^5 \log 5 \\\\ & \bbox[5px, border: 2px solid magenta] {-2} \end{split} \end{equation*}
Example 05
Simplify \(^6\log 12 + \: ^6\log 3\)
\begin{equation*} \begin{split} & ^6\log 12 + \: ^6\log 3 \\\\ & ^6\log(12 \:.\: 3) \\\\ & ^6\log 36 \\\\ & ^6\log 6^2\\\\ & 2 \:.\: ^6\log 6\\\\ & \bbox[5px, border: 2px solid magenta] {2} \end{split} \end{equation*}
Example 06
Simplify \(^2\log 100 - \: ^2\log 50\)
\begin{equation*} \begin{split} & ^2\log 100 - \: ^2\log 50 \\\\ & ^2 \log \left(\frac{100}{50} \right) \\\\ & ^2 \log 2 \\\\ & \bbox[5px, border: 2px solid magenta] {1} \end{split} \end{equation*}
Example 07
Simplify \(^2 \log 7 \:.\: ^7 \log 8\)
\begin{equation*} \begin{split} & ^2 \log \cancel{7} \:.\: ^{\cancel{7}} \log 8 \\\\ & ^2 \log 8 \\\\ & ^2 \log 2^3 \\\\ & 3 \:.\: ^2 \log 2 \\\\ & \bbox[5px, border: 2px solid magenta] {3} \end{split} \end{equation*}
Example 08
Simplify \((5)^{^{5}\log 7}\)
\begin{equation*} \begin{split} & ({\color {red} 5})^{^{{\color {red} 5}}\log {\color {blue} 7}} = \bbox[5px, border: 2px solid magenta] {7} \end{split} \end{equation*}
Exercise