Logarithm Laws

Similiar to the exponent laws and square root rules, logarithms have the logarithm laws.

1. The logarithm of a product

   $\log_b(u \cdot v) = \log_b(u) + \log_b(v)$

2. The logarithm of a bracket

   $\log_b(\frac{u}{v}) = \log_b(u) - \log_b(v)$

3. The logarithm of an exponent

   $\log_b(u^n)=n\cdot\log_b(u)$
   
   Roots can also be described as exponents.
   $\log_b(\sqrt[m]{u^n})$ $=\log_b(u^\frac{n}{m})$ $=\frac{n}{m}\cdot\log_b(u)$

More rules

• The change of base

  $\log_a(u) = \frac{\log_b(u)}{\log_b(a)}$

• The logarithm as an exponent

  $b^{\log_b(u)} = u$

Special cases

• $\log_b (b) = 1$
• $\log_b (1)=0$

Examples

• $\log_2 (32)=\log_2(4\cdot8)$ $=\log_2(4)+\log_2 (8)$ $=2+3=5$


• $\log_4 (\frac{1}{64})$ $=\log_4 (1)-\log_4 (64)$ $=0-3=-3$


• $\log_6 (36)=\log_6(6^2)$ $=2\cdot\log_6 (6)$ $=2\cdot1=2$


• $\log_{16} (64)$ $=\frac{\log_4(64)}{\log_4(16)}$ $=\frac{\log_4(4^3)}{\log_4(4^2)}$ $=\frac32$