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SPECIAL FUNCTIONS / STABLE LOG MAGNITUDE

Real Log Gamma Function Calculator

Evaluate ln Gamma(x) without first forming an overflowing Gamma value.

  • 01 Calculated in this tab
  • 02 Values stay in this browser tab
  • 03 Use boundary

Conversion input

Known value

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Function inputs

Preparing the calculator...

METHOD / WORKED EXAMPLE

Keep huge Gamma values finite by staying in log space

Log Gamma is useful when multiplying Gamma terms would overflow or lose scale. Its additive recurrence also makes ratios and likelihood expressions easier to audit numerically.

WORKED DEFAULT

Check the calculation with the default inputs

For x = 100, ln Gamma(100) is about 359.13420537 and log10 Gamma(100) is about 155.97000365, even though Gamma(100) itself is extraordinarily large.

  1. Enter a large argumentx = 100
  2. Evaluate in log spaceln Gamma(100) = 359.1342...
  3. Verify recurrenceln Gamma(101) - ln Gamma(100) = ln(100)

READ THE RESULT

Interpret the output in context

The logarithmic result communicates magnitude and supports stable subtraction or addition. Exponentiating it may still overflow, so downstream work should remain in log space whenever possible.

ASSUMPTIONS AND LIMITS

Know where the model stops

  • x is positive and real.
  • A logarithmic result is appropriate for the downstream calculation.
  • Natural and base-10 logarithms are not mixed without conversion.

Log Gamma preserves scale but not a direct Gamma value; do not compare the logarithm numerically with Gamma(x).

NIST DLMF: Gamma and log-Gamma graphics

COMMON QUESTIONS

Real Log Gamma Function Calculator FAQs

Why not calculate Gamma first and then take its logarithm?

Direct Gamma grows so quickly that it can overflow to infinity before the logarithm is applied. A log-Gamma algorithm evaluates the logarithmic expression itself, preserving a finite result across a much larger domain. This also improves stability when formulas add or subtract Gamma logarithms, as in beta functions, likelihoods, and combinatorial ratios.

What is the difference between ln Gamma and log10 Gamma?

They represent the same positive Gamma magnitude using different logarithm bases. Natural log is standard in calculus, probability, and numerical formulas; base-10 log is convenient for interpreting decimal order of magnitude. Convert with log10 Gamma = ln Gamma divided by ln(10), and never add values expressed in different bases without conversion.

Can I recover Gamma by exponentiating this result?

Only when the result is small enough that exp(log Gamma) fits the numeric range. For large arguments, exponentiation overflows even though log Gamma remains finite. Often recovery is unnecessary: ratios can be computed by subtracting logs, and products by adding them. Preserve log space until a final representable quantity is genuinely required.

Use boundary

Calculation path

Evaluate the logarithmic Lanczos form directly, then check the additive recurrence without exponentiating the potentially enormous result.

Calculation path

ln Gamma(x + 1) = ln(x) + ln Gamma(x).