Logarithm Calculator - Calculate Log, ln, log10
Calculate logarithms with any base using our free online calculator. Supports natural log (ln), common log (log10), binary log (log2), and custom bases with detailed explanations.
Free Logarithm Calculator - Calculate Any Log Function Online
Calculate logarithms with any base using our comprehensive online calculator. Whether you need natural logarithm (ln), common logarithm (log₁₀), binary logarithm (log₂), or custom base logarithms, our tool provides accurate results with detailed explanations.
Perfect for students, engineers, scientists, and anyone working with exponential relationships, pH calculations, decibel measurements, or data analysis. Our calculator handles all logarithmic functions with precision and provides step-by-step breakdowns for learning.
Calculate Logarithms
Base must be positive and not equal to 1
Enter the number you want to find the logarithm of (must be positive)
Understanding Logarithmic Functions
How to Calculate Logarithms
Step 1: Identify the Base
Determine which logarithm you need: ln (base e), log (base 10), log₂ (base 2), or a custom base.
Step 2: Enter the Number
Input the positive number you want to find the logarithm of.
Step 3: Calculate & Verify
Our calculator computes the result and shows verification: base^result = number.
Common Logarithm Examples
Natural Logarithm
ln(e) = 1, ln(1) = 0, ln(e²) = 2
Common Logarithm
log(10) = 1, log(100) = 2, log(1000) = 3
Binary Logarithm
log₂(2) = 1, log₂(4) = 2, log₂(8) = 3
Common Logarithm Values Reference Chart
Quick reference for frequently used logarithm values across different bases.
Natural Logarithm (ln)
Base: e ≈ 2.718
| x | ln(x) |
|---|---|
| 1 | 0 |
| e | 1 |
| 2 | 0.693 |
| 3 | 1.099 |
| 5 | 1.609 |
| 10 | 2.303 |
| e² | 2 |
Common Logarithm (log)
Base: 10
| x | log(x) |
|---|---|
| 1 | 0 |
| 2 | 0.301 |
| 5 | 0.699 |
| 10 | 1 |
| 100 | 2 |
| 1000 | 3 |
| 10000 | 4 |
Binary Logarithm (log₂)
Base: 2
| x | log₂(x) |
|---|---|
| 1 | 0 |
| 2 | 1 |
| 4 | 2 |
| 8 | 3 |
| 16 | 4 |
| 32 | 5 |
| 64 | 6 |
Real-World Applications of Logarithms
pH Scale
pH = -log₁₀[H⁺]. Measures acidity/alkalinity from 0-14. Each unit represents a 10x change in hydrogen ion concentration.
Decibel Scale
dB = 10 × log₁₀(P₁/P₀). Measures sound intensity on a logarithmic scale. Every 10 dB increase represents 10x more power.
Richter Scale
Earthquake magnitude using log₁₀ of seismic wave amplitude. Each whole number increase means 10x more powerful earthquake.
Computer Science
Algorithm complexity analysis, binary search trees, data compression. log₂(n) appears frequently in computational efficiency.
Finance
Compound interest calculations, investment growth modeling, and determining time needed to reach financial goals.
Psychology
Weber-Fechner law describes perception intensity using logarithms. Human senses respond logarithmically to stimulus changes.
Tips for Working with Logarithms
Common Mistakes to Avoid
Negative Numbers
Logarithms are only defined for positive real numbers. log(0) and log(negative) are undefined.
Base Restrictions
The base must be positive and not equal to 1. log₁(x) is undefined because 1ʸ = 1 for all y.
Property Confusion
Remember: log(a + b) ≠ log(a) + log(b). Use log(ab) = log(a) + log(b) for multiplication.
Helpful Strategies
Use Change of Base
Convert any logarithm to natural or common logs: log_b(x) = ln(x)/ln(b) = log(x)/log(b)
Memorize Key Values
Know that log_b(1) = 0, log_b(b) = 1, and log_b(b^n) = n for any valid base b.
Practice Properties
Master the product, quotient, and power rules. They're essential for simplifying complex expressions.