Section 7.2: Laws of Logarithms
This section covers the key logarithmic properties: the product, quotient, and power laws, and demonstrates how to simplify logarithmic expressions using these laws.
Example 1: Product Law
Simplify \( \log_2 8 + \log_2 4 \).
Step 1: Apply product law: \( \log_2 8 + \log_2 4 = \log_2 (8 \cdot 4) \)
Step 2: Multiply: \( \log_2 32 \)
Step 3: Evaluate: \( \log_2 32 = 5 \)
Example 2: Quotient Law
Simplify \( \log_5 125 - \log_5 25 \).
Step 1: Apply quotient law: \( \log_5 125 - \log_5 25 = \log_5 (125 / 25) \)
Step 2: Simplify: \( \log_5 5 \)
Step 3: Evaluate: \( \log_5 5 = 1 \)
Example 3: Power Law
Simplify \( \log_3 (27^2) \).
Step 1: Apply power law: \( \log_3 (27^2) = 2 \cdot \log_3 27 \)
Step 2: Evaluate \( \log_3 27 = 3 \)
Step 3: Multiply: \( 2 \cdot 3 = 6 \)
Practice Problems
- Simplify \( \log_2 16 + \log_2 8 \)
- Simplify \( \log_10 1000 - \log_10 10 \)
- Simplify \( \log_4 (64^3) \)
- Simplify \( \log_3 9 + \log_3 27 \)
- Simplify \( \log_5 25^2 - \log_5 5 \)