Logarithmic Rule

$^{a} \log b = c \to a^{c} = b$

$a > 0, b > 0$ dan $a \neq 1$

Example 01

Find the value of $x$ from $^{2} \log x = 3$

$\begin{aligned} ^{2} \log x = 3 \\ x = 2^{3} \\ x = 8 \end{aligned}$

Exponent

$^{a} \log c^{p} = p .^{a} \log c$

$^{a^{q}} \log c = \frac{1}{q} .^{a} \log c$

$^{a^{q}} \log c^{p} = \frac{p}{q} .^{a} \log c$

Sum and Substraction

$^{c} \log a +^{c} \log b =^{c} \log a . b$

$^{c} \log a -^{c} \log b =^{c} \log \frac{a}{b}$

Multiplication

$^{a} \log b .^{b} \log c =^{a} \log c$

Reciprocal

$^{a} \log b = \frac{1}{^{b} \log a}$

Exponent Form

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

Example 02

$\begin{aligned} ^{7} \log \frac{1}{49} \\ ^{7} \log 7^{- 2} \\ - 2 .^{7} \log 7 \\ - 2 \end{aligned}$

Example 03

$\begin{aligned} ^{8} \log 2 \\ ^{2^{3}} \log 2 \\ \frac{1}{3} .^{2} \log 2 \\ \frac{1}{3} \end{aligned}$

Example 04

$\begin{aligned} ^{25} \log \frac{1}{625} \\ ^{5^{2}} \log 5^{- 4} \\ \frac{- 4}{2} .^{5} \log 5 \\ - 2 \end{aligned}$

Example 05

Simplify $^{6} \log 12 +^{6} \log 3$

$\begin{aligned} ^{6} \log 12 +^{6} \log 3 \\ ^{6} \log (12 . 3) \\ ^{6} \log 36 \\ ^{6} \log 6^{2} \\ 2 .^{6} \log 6 \\ 2 \end{aligned}$

Example 06

Simplify $^{2} \log 100 -^{2} \log 50$

$\begin{aligned} ^{2} \log 100 -^{2} \log 50 \\ ^{2} \log (\frac{100}{50}) \\ ^{2} \log 2 \\ 1 \end{aligned}$

Example 07

Simplify $^{2} \log 7 .^{7} \log 8$

$\begin{aligned} ^{2} \log 7 .^{7} \log 8 \\ ^{2} \log 8 \\ ^{2} \log 2^{3} \\ 3 .^{2} \log 2 \\ 3 \end{aligned}$

Example 08

Simplify $(5)^{^{5} \log 7}$

$\begin{aligned} (5)^{^{5} \log 7} = 7 \end{aligned}$