Laws of logarithms
Product, quotient and power laws of logarithms.
Indices & Logarithms · O/L Maths
Level 1: The product law
Use log a + log b = log (ab)
Multiplying becomes adding logarithms.
log_b (x y) = log_b x + log_b y
Why? bᵐ × bⁿ = b^m + n: indices add, and logarithms are indices.
- log₂ 4 + log₂ 8
- Product law: log₂ (4 × 8)
- = log₂ 32 = 5
Level 2: The quotient law
Use log a − log b = log (a ÷ b)
Dividing becomes subtracting logarithms.
log_b (x ÷ y) = log_b x − log_b y
- log₁₀ 100000 − log₁₀ 100
- = log₁₀ (100000 ÷ 100) = log₁₀ 1000
- = 3
Level 3: The power law
Use log aⁿ = n log a
The index comes down in front as a multiplier.
log_b (xⁿ) = n log_b x
Why? xⁿ = x × x × … × x (n times), so the logs add n times.
- log₂ 4³
- Power law: 3 × log₂ 4
- = 3 × 2 = 6
Level 4: Combining the laws
Use several laws in one problem
To solve a problem: combine with the product, quotient and power laws into one logarithm, then ask "which power?".
- 2 log₁₀ 5 + log₁₀ 4
- Power law: 2 log₁₀ 5 = log₁₀ 25
- Product law: log₁₀ 25 + log₁₀ 4 = log₁₀ 100
- = 2