Enter a positive number greater than 0.
The argument of the logarithm — must be > 0
Select the base or enter a custom value below
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Sources & Methodology
Logarithm formulas verified against Khan Academy mathematics curriculum and the NIST Digital Library of Mathematical Functions (DLMF).
Khan Academy — Introduction to Logarithms
Standard curriculum reference for logarithm definitions, properties, and the change of base formula used in this calculator
NIST Digital Library of Mathematical Functions — Logarithm Functions
Authoritative mathematical reference for logarithm identities and the relationship between common log, natural log, and arbitrary base logarithms
Methodology: logₙ(x) computed using the change of base formula: logₙ(x) = ln(x) ÷ ln(b). JavaScript Math.log() returns the natural log. Common log: Math.log(x)/Math.LN10. Binary log: Math.log(x)/Math.LN2. Custom base b: Math.log(x)/Math.log(b). Verification: b^result should equal x within floating-point precision.
⏱ Last reviewed: March 2026
How to Calculate Logarithms
A logarithm answers the question: “to what power must the base be raised to produce this number?” Written as logₙ(x) = y, it means b^y = x. For example, log₁₀(1000) = 3 because 10³ = 1000. Logarithms are the inverse of exponential functions and appear throughout mathematics, science, engineering, and finance.
The Three Standard Logarithms
Common log: log(x) = log₁₀(x) = ln(x) ÷ ln(10)
Example: log(1000) = 3 ✔ Verify: 10³ = 1000
log(100) = 2 log(10) = 1 log(1) = 0 log(0.1) = −1
log(100) = 2 log(10) = 1 log(1) = 0 log(0.1) = −1
Natural log: ln(x) = logₑ(x) e ≈ 2.71828
Example: ln(e) = 1 ln(e²) = 2 ln(1) = 0 ln(10) ≈ 2.3026
ln(x) = log(x) × ln(10) ≈ log(x) × 2.3026
ln(x) = log(x) × ln(10) ≈ log(x) × 2.3026
Change of Base: logₙ(x) = ln(x) ÷ ln(b) = log(x) ÷ log(b)
Example: log₂(64) = ln(64) ÷ ln(2) = 4.1589 ÷ 0.6931 = 6
Verify: 2⁶ = 64 ✔
Verify: 2⁶ = 64 ✔
Common Logarithm Reference Table
| x | log₁₀(x) | ln(x) | log₂(x) | Verify |
|---|---|---|---|---|
| 1 | 0.0000 | 0.0000 | 0.0000 | 10⁰ = 1 |
| 2 | 0.3010 | 0.6931 | 1.0000 | 2¹ = 2 |
| 10 | 1.0000 | 2.3026 | 3.3219 | 10¹ = 10 |
| 100 | 2.0000 | 4.6052 | 6.6439 | 10² = 100 |
| 1,000 | 3.0000 | 6.9078 | 9.9658 | 10³ = 1,000 |
| 1,000,000 | 6.0000 | 13.8155 | 19.9316 | 10⁶ = 1,000,000 |
The Five Logarithm Rules
- Product Rule: logₙ(x·y) = logₙ(x) + logₙ(y)
- Quotient Rule: logₙ(x/y) = logₙ(x) − logₙ(y)
- Power Rule: logₙ(xⁿ) = n × logₙ(x)
- Identity: logₙ(b) = 1 and logₙ(1) = 0
- Change of Base: logₙ(x) = log(x) ÷ log(b)
💡 Real-World Uses: The Richter scale for earthquakes is base-10 logarithmic — each unit increase represents a 10× stronger quake. pH is −log₁₀[H⁺], so pH 4 is 10× more acidic than pH 5. Compound interest doubling time = ln(2) ÷ r ≈ 0.693 ÷ r. Binary log tells how many bits are needed: log₂(256) = 8 bits.
Frequently Asked Questions
A logarithm answers: to what power must the base be raised to get this number? logₙ(x) = y means b^y = x. For example, log₁₀(1000) = 3 because 10³ = 1000. The three most common logarithms are common log (base 10), natural log (base e ≈ 2.71828), and binary log (base 2). Logarithms are the inverse operation of exponentiation.
Log base 10, written log(x), equals ln(x) ÷ ln(10). On a scientific calculator, press the LOG button. Examples: log(10) = 1, log(100) = 2, log(1000) = 3, log(1) = 0, log(0.1) = −1. The result tells you how many times you multiply 10 by itself to reach that number. For non-powers-of-10, the result is a decimal.
The natural log ln(x) uses base e (Euler’s number ≈ 2.71828). On a calculator, press LN. Examples: ln(1) = 0, ln(e) = 1, ln(e²) = 2, ln(10) ≈ 2.3026. The natural log is used in calculus derivatives, compound interest formulas, and exponential growth and decay. The relationship: ln(x) = log(x) × 2.3026.
Change of Base: logₙ(x) = log(x) ÷ log(b) = ln(x) ÷ ln(b). This lets you calculate any base using a standard calculator. Example: log₂(64) = log(64) ÷ log(2) = 1.80618 ÷ 0.30103 = 6. Verify: 2⁶ = 64. You can use either common log or natural log — the ratio is always the same.
Log base 2 (binary logarithm, log₂) is fundamental in computer science. It tells you how many bits are needed: log₂(256) = 8 bits. It appears in algorithm complexity analysis (Big O notation), data compression, binary search, and information theory. log₂(x) = ln(x) ÷ ln(2) ≈ ln(x) ÷ 0.6931.
The five key rules: 1) Product: logₙ(xy) = logₙ(x) + logₙ(y). 2) Quotient: logₙ(x/y) = logₙ(x) − logₙ(y). 3) Power: logₙ(xⁿ) = n × logₙ(x). 4) Identity: logₙ(b) = 1 and logₙ(1) = 0. 5) Change of Base: logₙ(x) = log(x) ÷ log(b). These rules let you simplify complex log expressions and solve log equations.
log (common logarithm) uses base 10: log(1000) = 3. ln (natural logarithm) uses base e ≈ 2.71828: ln(e³) = 3. They are related by ln(x) = log(x) × 2.3026. In mathematics and physics, ln is the natural default. In engineering and everyday use, log base 10 is standard. On calculators, LOG means base 10 and LN means base e.
No. The argument x in logₙ(x) must be positive (x > 0). The base b must also be positive and not equal to 1. No real power of a positive base can produce zero or a negative number. log(0) and log(−5) are undefined in real numbers. In complex number theory, log(−x) involves imaginary components, but that is outside standard calculator scope.
To solve logₙ(x) = y for x: raise b to both sides: x = b^y. Example: log₁₀(x) = 3 means x = 10³ = 1000. To solve log(2x) = 2: 2x = 10² = 100, so x = 50. For equations with multiple log terms, use log rules to combine them first. Always verify that your solution makes the original argument positive.
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