Logarithm Calculator
log_b(x) for any base b > 0, b ≠ 1, x > 0. Common bases shown side-by-side.
Result
- log₁₀(x)2.000000
- ln(x)4.605170
- log₂(x)6.643856
- Change-of-baselog_10(x) = ln(x) / ln(10)
Step-by-step
- Use change-of-base: log_10(100) = ln(100) / ln(10) = 4.6052 / 2.3026 = 2.00000000.
How to use this calculator
- Enter x and the base.
- Read the log; the breakdown shows base-10, natural, and base-2 logs for reference.
About this calculator
A logarithm answers: "to what power must I raise the base to get x?" log₁₀(100) = 2 because 10² = 100. Common bases: base 10 (log), base e ≈ 2.718 (ln, natural log), base 2 (used in computer science). Change-of-base formula lets you compute log in any base from natural log: log_b(x) = ln(x) / ln(b).
How it works — the formula
logₐ(x) = y ⟺ aʸ = x (general)
log(x) := log₁₀(x) (common log)
ln(x) := logₑ(x), e ≈ 2.71828 (natural log)
Change of base: logₐ(x) = ln(x) / ln(a)The logarithm logₐ(x) is the exponent y to which the base a must be raised to produce x. The base must be positive and not 1; the argument must be strictly positive. The common log uses base 10 (standard for orders of magnitude — pH, decibels, Richter scale), and the natural log uses base e (standard for continuous-rate models — radioactive decay, compound interest, population growth). The change-of-base identity lets any calculator with one log function compute any other.
Worked examples
- Inputs:
- log₁₀(1000)
- Output:
- 3 (because 10³ = 1000)
- Inputs:
- ln(e²)
- Output:
- 2 (because eˣ and ln are inverses)
- Inputs:
- log₂(10) via natural log
- Output:
- log₂(10) = ln(10) / ln(2) ≈ 3.3219
Limitations
- log of zero is −∞ as a limit; log of a negative number is complex (not real).
- Floating-point representation has ~15–16 significant digits; logarithms of very large or very small numbers may lose precision in the trailing digits.
- Different mathematical communities use "log" to mean either log₁₀ (most science/engineering) or logₑ (most pure math) — clarify the base when ambiguous.
- The series ln(1 + x) = x − x²/2 + x³/3 − … converges only for −1 < x ≤ 1; for larger arguments, use the change-of-base identity or built-in implementations.
Computed with the platform's IEEE 754 Math.log family; agreement with NIST tabulated values exceeds 12 significant digits across the typical input range.