4.4: Laws of Logarithms

Motivation

Logarithms are exponents.

The three exponent laws \[x^A \cdot x^B = x^{A + B} \qquad \qquad \dfrac{x^A}{x^B} = x^{A-B} \qquad \qquad (x^A)^C = a^{A\cdot C} \] have corresponding forms in logarithms.

Laws of Logarithms

Laws of Logarithms
Let $a > 0, a \neq 1$ and $A,B,C \in \mathbb{R}$ where $A > 0$ and $B > 0$.
  1. $\log_a(AB) = \log_a A + \log_a B$
  2. $\log_a\left(\dfrac{A}{B}\right) = \log_a A - \log_a B$
  3. $\log_a(A^C) = C\cdot\log_a A$
Evaluate each expression using Laws of Logarithms.
  1. $\log_8 16 + \log_8 4$
  2. $\log_2 56 - \log_2 7$
  3. $-\dfrac{2}{3}\ln\left(\dfrac{1}{e^3}\right)$

These laws are used in Calculus. For example:

Expand the expression $\ln\big(x(x-1)\big)^{x^2 - 1}.$

Combining logarithms means to convert terms into one logarithm.

Combine the expression $3\ln (x-1) + \dfrac{1}{2}\ln(x + 1) - 3\ln(x^2 + 1).$