Math Logarithms Logarithm laws

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

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

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

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

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