Logarithm Calculator
Calculate natural log (ln), common log, and custom base logarithms
💡 Share this logarithm calculation with others
📚 Examples, Rules & Help
⚡Quick Examples - Try These Calculations
🔍How It Works
Logarithm Definition
Basic Definition:
Properties & Rules
Essential Properties:
Applications
Common Uses:
🌍Real-World Applications
❓Frequently Asked Questions
What's the difference between ln and log?
ln (natural logarithm): Uses base e ≈ 2.71828
log (common logarithm): Usually means base 10
In calculus and science, ln is more common. In engineering, log₁₀ is often used.
Example: ln(e) = 1, log(10) = 1
Why can't you take the log of negative numbers?
In real numbers, logarithms are only defined for positive values.
This is because no real power of a positive base can equal a negative number.
Example: There's no real x such that 10^x = -1
Complex numbers extend this, but that's beyond basic logarithms.
How do I convert between different logarithm bases?
Use the change of base formula:
log_b(x) = ln(x) / ln(b)
Or using common logarithms:
log_b(x) = log₁₀(x) / log₁₀(b)
Example: log₂(8) = ln(8) / ln(2) = 3
What are logarithms used for in real life?
Science: pH levels, earthquake magnitudes, sound levels
Finance: Compound interest calculations, investment growth
Technology: Computer algorithms, data compression
Psychology: Human perception (Weber-Fechner law)
🎯Common Use Cases
🎓 Academic Learning
- • Algebra and precalculus homework
- • Calculus derivative and integral practice
- • Chemistry pH calculations
- • Physics exponential decay problems
🔬 Scientific Research
- • pH and chemical concentration
- • Radioactive decay analysis
- • Earthquake magnitude calculations
- • Biological growth modeling
💼 Engineering Applications
- • Signal processing and filtering
- • Decibel and sound level measurements
- • Control system design
- • Information theory calculations
💰 Financial Analysis
- • Compound interest calculations
- • Investment growth analysis
- • Economic modeling and forecasting
- • Risk assessment and probability