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Understanding Logarithms
MathLearn what logarithms are and how to use them.
What is a Logarithm?
A logarithm is the inverse of exponentiation. If you know that bx = y, then logb(y) = x. In simple terms, a logarithm answers: "To what power must I raise the base to get this number?"
The Two Common Logarithms
Common Logarithm (log₁₀ or just "log")
Uses base 10. Commonly used in chemistry (pH), earthquake measurement (Richter scale), and sound (decibels).
- log(10) = 1 (because 10¹ = 10)
- log(100) = 2 (because 10² = 100)
- log(1000) = 3 (because 10³ = 1000)
Natural Logarithm (ln)
Uses base e ≈ 2.71828. Essential in calculus, physics, and exponential growth/decay problems.
- ln(e) = 1
- ln(1) = 0
- ln(e²) = 2
Key Logarithm Rules
| Rule | Formula |
|---|---|
| Product Rule | log(ab) = log(a) + log(b) |
| Quotient Rule | log(a/b) = log(a) - log(b) |
| Power Rule | log(aⁿ) = n × log(a) |
| Change of Base | log_b(x) = log(x) / log(b) |
Reference Table
Check out our complete Logarithm Table for values from 1 to 100!