Compute log₁₀, ln, log₂, or any custom base instantly
Input
Must be > 0
Must be > 0 and ≠ 1
Results
3
log₁₀(1000) = 3
6.907755279
ln(1000) = 6.907755279
9.965784285
log₂(1000) = 9.965784285
4.292029674
log_5(1000) = ln(1000) / ln(5) = 4.292029674
Logarithm Identities
| Identity | Description |
|---|---|
| logₙ(1) = 0 | Log of 1 is always 0 for any base |
| logₙ(n) = 1 | Log of the base itself is always 1 |
| logₙ(a·b) = logₙ(a) + logₙ(b) | Product rule |
| logₙ(a/b) = logₙ(a) − logₙ(b) | Quotient rule |
| logₙ(aᵖ) = p · logₙ(a) | Power rule |
| log_b(x) = ln(x) / ln(b) | Change of base formula |
The logarithm calculator lets you compute the log of any positive number for any base — including log base 10 (common logarithm), the natural logarithm (ln, base e), the binary logarithm (log₂), or a completely custom base you specify. Results are shown instantly with the full formula and adjustable significant figures, making it ideal for students, engineers, and anyone working with exponential relationships.
log (or log₁₀) is the common logarithm with base 10, widely used in science and engineering. ln is the natural logarithm with base e (≈ 2.71828), fundamental in calculus and continuous growth models. log₂ is the binary logarithm with base 2, essential in computer science for measuring information in bits.
Select 'All common + custom base' mode, enter your number and the desired base. The calculator uses the change-of-base formula: log_b(x) = ln(x) / ln(b), which works for any valid base greater than 0 and not equal to 1.
Logarithms are only defined for positive real numbers. log(0) approaches negative infinity, and logarithms of negative numbers require complex numbers, which are outside the scope of a standard real-valued calculator.
Significant figures control how many meaningful digits appear in the result. For example, at 4 significant figures, log₁₀(1000) = 3.000; at 10 figures it shows more decimal places for irrational results like log₁₀(2) = 0.3010299957.