Logaritma

\(^a\log b = c \rightarrow a^c = b\)

\(a > 0, \: b > 0\) dan \(a \neq 1\)

Contoh 01

Tentukan nilai \(x\) dari \(^2 \log x = 3\)

\begin{equation*} \begin{split} & ^2 \log x = 3 \\\\ & x = 2^3 \\\\ & \bbox[5px, border: 2px solid magenta] {x = 8} \end{split} \end{equation*}

Sifat Eksponen

\(^a \log c^p = p \:.\: ^a \log c\)

\(^{a^q} \log c = \dfrac{1}{q} \:.\: ^a \log c\)

\(^{a^q} \log c^p = \dfrac{p}{q} \:.\: ^a \log c\)

Sifat penjumlahan dan Pengurangan logaritma

\(^c\log a + \: ^c\log b  = \: ^c \log a \:.\: b\)

\(^c\log a - \: ^c\log b  = \: ^c \log \dfrac{a}{b}\)

Sifat Perkalian logaritma

\(^a\log b \:.\: ^b\log c  = \: ^a \log c\)

Bentuk kebalikan

\(^a\log b = \dfrac{1}{^b \log a}\)

Bentuk pangkat

\((c)^{^{c}\log a} = a\)

Contoh 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*}

Contoh 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*}

Contoh 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*}

Contoh 05

Sederhanakan bentuk logaritma dari \(^6\log 12 + \: ^6\log 3\)

\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*}

Contoh 06

Sederhanakan bentuk logaritma dari \(^2\log 100 - \: ^2\log 50\)

\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*}

Contoh 07

Sederhanakan bentuk \(^2 \log 7 \:.\: ^7 \log 8\)

\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*}

Contoh 08

Sederhanakan bentuk \((5)^{^{5}\log 7}\)

\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*}