What is a Logarithm?
A logarithm answers the question: "To what exponent must we raise the base to get this number?" If b^y = x, then log_b(x) = y.
log₁₀(100) = 2 (because 10² = 100)
ln(e) = 1 (because e¹ = e)
log₂(8) = 3 (because 2³ = 8)
ln(e) = 1 (because e¹ = e)
log₂(8) = 3 (because 2³ = 8)
Types of Logarithms
- Common Log (log₁₀): Base 10, widely used in science and engineering.
- Natural Log (ln): Base e (≈ 2.71828), used in calculus and natural growth models.
- Binary Log (log₂): Base 2, used in computer science and information theory.
Logarithm Properties — Complete Rules
| Property | Formula | Example (log₁₀) |
|---|---|---|
| Product Rule | log(xy) = log(x) + log(y) | log(100) = log(10)+log(10) = 2 |
| Quotient Rule | log(x/y) = log(x) − log(y) | log(100/10) = 2−1 = 1 |
| Power Rule | log(xⁿ) = n·log(x) | log(1000) = 3·log(10) = 3 |
| Change of Base | log_a(x) = ln(x)/ln(a) | log₂(8) = ln(8)/ln(2) = 3 |
| log of 1 | log_b(1) = 0 | log(1) = 0 |
| log of base | log_b(b) = 1 | log₁₀(10) = 1 |
Common Logarithm Values
log₁₀(1) = 0 log₁₀(10) = 1 log₁₀(100) = 2
log₁₀(1000) = 3 log₁₀(0.1) = −1
ln(1) = 0 ln(e) = 1 ln(e²) = 2
log₂(1) = 0 log₂(2) = 1 log₂(8) = 3 log₂(1024) = 10
log₁₀(1000) = 3 log₁₀(0.1) = −1
ln(1) = 0 ln(e) = 1 ln(e²) = 2
log₂(1) = 0 log₂(2) = 1 log₂(8) = 3 log₂(1024) = 10
Real-World Applications
- Sound (Decibels): Sound level (dB) = 10 × log₁₀(I/I₀). Every 10 dB increase = 10× louder.
- Earthquakes (Richter scale): Each unit increase = 10× more ground motion amplitude.
- pH scale: pH = −log₁₀[H⁺]. Each pH unit = 10× change in acidity.
- Finance: ln used in continuous compound interest: A = Pe^(rt)
- Computer science: log₂ in binary search, tree height, bit complexity analysis
- Information theory: Shannon entropy uses log₂ to measure information in bits