🎓 Mathematics • Algebra
100% Client-Side Privacy
Logarithm Calculator
Computes common logarithms (base 10), natural logarithms (base e), binary logarithms (base 2), and arbitrary base logs.
Logarithm Calculator: log_b(x)
Calculate natural log (ln), common log (log₁₀), binary log (log₂), or any base.
Presets:
at Base
Result: log10(100)
2
Natural Log (ln / Base e)ln(100) = 4.60517
Common Log (log₁₀ / Base 10)log₁₀(100) = 2
Binary Log (log₂ / Base 2)log₂(100) = 6.643856
Step-by-Step Change of Base
Change of Base Formula: log_10(100) = ln(100) / ln(10) = 4.60517 / 2.30259 = 2
Logarithm Laws Quick Reference
Product: log_b(xy) = log_b(x) + log_b(y)
Quotient: log_b(x/y) = log_b(x) - log_b(y)
Power: log_b(xᵏ) = k · log_b(x)
Base Identity: log_b(b) = 1, log_b(1) = 0
What is a Logarithm?
A logarithm is the inverse operation of exponentiation, determining what power produces a given value.
Logarithms compress large dynamic ranges into readable scales (e.g. Richter scale, decibels, pH).
Formula & Step-by-Step Calculation
log_b(x) = ln(x) / ln(b)
Change of base formula for calculating arbitrary base logarithms.
Worked Step-by-Step Examples
Example 1
Find log₁₀(1000)
Solution: 3
• 10³ = 1000 ⟹ log₁₀(1000) = 3
Common Real-World & Academic Use Cases
- ✓ Sound intensity decibels
- ✓ Chemistry pH calculations
- ✓ Computer science algorithm complexity
How to Use the Logarithm Calculator
1
Enter Number & Base
Input positive number and base.
2
Read Logarithm
Inspect natural and common log values.
Frequently Asked Questions
Q: What is the natural log ln?
Logarithm with base e (Euler’s number e ≈ 2.71828).