# Logarithmic and Exponential Functions

### Logarithmic Rule

###### 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*}