Logarithm Calculator
Common Logarithm Values
| Value | log₁₀ | ln (log_e) | log₂ |
|---|---|---|---|
| 1 | 0 | 0 | 0 |
| 2 | 0.301 | 0.693 | 1 |
| e | 0.434 | 1 | 1.443 |
| 10 | 1 | 2.303 | 3.322 |
| 100 | 2 | 4.605 | 6.644 |
| 1000 | 3 | 6.908 | 9.966 |
Log Rules
Related Calculators
Understanding Logarithms
What is a Logarithm?
A logarithm is the inverse operation of exponentiation. If bˣ = y, then logb(y) = x. In simple terms, the logarithm answers the question: "To what power must we raise the base to get this number?"
Basic Definition
Example: log₁₀(100) = 2 because 10² = 100
Common Logarithm Types
Common Log (log)
Base 10 logarithm
Natural Log (ln)
Base e (≈2.718) logarithm
Binary Log (log₂)
Base 2 logarithm
Logarithm Properties
| Property | Formula | Example |
|---|---|---|
| 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(10/100) = 1 - 2 = -1 |
| Power Rule | log(xⁿ) = n × log(x) | log(10³) = 3 × log(10) = 3 |
| Change of Base | logₐ(x) = logb(x) / logb(a) | log₂(8) = log(8) / log(2) |
| Identity | logₐ(a) = 1 | log₁₀(10) = 1 |
Antilogarithm
The antilogarithm (or inverse logarithm) is the reverse operation of a logarithm. If log(x) = y, then antilog(y) = x = 10ʸ (for common logs).
Antilog Definition
Change of Base Formula
The change of base formula allows you to convert between different logarithm bases. This is useful when your calculator only has log₁₀ and ln buttons.
Example: log₂(8) = log(8) / log(2) = 0.903 / 0.301 = 3
Real-World Applications
Science & Engineering
- • pH scale (acidity/alkalinity)
- • Decibels (sound intensity)
- • Richter scale (earthquakes)
- • Radioactive decay
Finance & Computing
- • Compound interest calculations
- • Algorithm complexity (Big O)
- • Information theory (bits)
- • Data compression
The Number e
The number e (≈2.71828) is called Euler's number and is the base of natural logarithms. It appears naturally in growth and decay problems, compound interest with continuous compounding, and many areas of calculus. The natural logarithm ln(x) = logₑ(x) is particularly important in mathematics and science.
Frequently Asked Questions
What is an antilogarithm?
An antilogarithm (antilog) is the reverse of a logarithm. If log base b of x equals y, then antilog base b of y equals x, which is b raised to the power y. For common logs: antilog(2) = 10 squared = 100. Essentially, antilog 'undoes' what log does.
What is the natural logarithm e and why is it special?
The natural logarithm uses base e (approximately 2.71828), called Euler's number. It's special because it appears naturally in continuous growth and decay, compound interest, and calculus. The derivative of e to the x is itself, making it fundamental in mathematics.
How do you solve log equations?
Convert between log and exponential forms. If log base b of x = y, then b to the y = x. For example, if log base 2 of x = 5, then x = 2 to the 5 = 32. Use log properties to combine or separate terms: log(ab) = log(a) + log(b).
Why are logarithms used for the Richter scale?
The Richter scale uses base-10 logarithms because earthquake energy spans enormous ranges. A magnitude 8 earthquake releases about 32 times more energy than magnitude 7. Using logs, each whole number increase represents a 10x increase in amplitude and about 31.6x more energy.