# Section 4.4 - Properties of Logarithms

Section Objectives

1. Apply the logarithm laws.
2. Use the logarithm laws to expand logarithmic expressions.
3. Use the logarithm laws to expand logarithmic expressions.
4. Apply the change-of-base formula.

### Logarithm Laws

The following properties of logarithms are very useful.

• $\log_b 1 = 0$
• $\log_b b = 1$
• $\log_b b^p = p$
• $b^{\log_b x} = x$
• $\log_b (xy)= \log_b x + \log_b y$
• $\displaystyle \log_b \left( \frac{x}{y} \right) = \log_b x - \log_b y$
• $\log_b x^p = p \log_b x$

Warning: Do not apply the logarithm laws when they do not apply. For example,

#### Examples

• Expand and simplify: $\quad \log_2 24 = \log_2 (8 \cdot 3)$

• Expand and simplify: $\quad \displaystyle \ln \left( \frac{e}{12} \right)$

• Expand and simplify: $\quad \displaystyle \ln \left( \frac{z^3}{xy^5} \right)$

• Condense to a single logarithm: $\quad \frac{1}{2} \log x + 3 \log (x+1)$

• Condense to a single logarithm: $\quad 2[\ln(x+3)-2\ln(x-2)]$

### Change-of-Base Formula

For any valid bases $a$ and $b$ and any positive real number $x$,

The change of base-of-base formula can be used to convert any log to a common or natural log.

• Compute $\log_2 100$ by using the change-of-base formula.