Lesson 64: Laws of Logarithms
Logarithms follow specific rules that make it easier to simplify and solve logarithmic expressions.
Key Laws of Logarithms
- Product Rule: log_b(M × N) = log_b(M) + log_b(N)
- Quotient Rule: log_b(M ÷ N) = log_b(M) − log_b(N)
- Power Rule: log_b(M^p) = p × log_b(M)
- Change of Base Formula: log_b(M) = log_k(M) ÷ log_k(b), for any positive base k ≠1
Examples
Example 1 (Product Rule): log_2(8 × 4) = log_2(8) + log_2(4) = 3 + 2 = 5
Example 2 (Quotient Rule): log_5(125 ÷ 5) = log_5(125) − log_5(5) = 3 − 1 = 2
Example 3 (Power Rule): log_3(9^2) = 2 × log_3(9) = 2 × 2 = 4
Example 4 (Change of Base): log_2(32) = log_10(32) ÷ log_10(2) ≈ 5
Practice Problems
- log_2(16 × 8)
- log_7(49 ÷ 7)
- log_5(25^3)
- log_10(1000) using log_2 (change of base formula)
- Simplify: log_3(27 × 9) − log_3(3)