In mathematics, the natural logarithm of 2 is the unique real number argument such that the exponential function equals two.

It appears regularly in various formulas and is also given by the alternating harmonic series.

The decimal value of the natural logarithm of 2 (sequence A002162 in the OEIS) truncated at 30 decimal places is given by: The logarithm of 2 in other bases is obtained with the formula The common logarithm in particular is (OEIS: A007524) The inverse of this number is the binary logarithm of 10: By the Lindemann–Weierstrass theorem, the natural logarithm of any natural number other than 0 and 1 (more generally, of any positive algebraic number other than 1) is a transcendental number.

(γ is the Euler–Mascheroni constant and ζ Riemann's zeta function.)

gives: The natural logarithm of 2 occurs frequently as the result of integration.

The logarithm of 2 is useful in the sense that the powers of 2 are rather densely distributed; finding powers 2i close to powers bj of other numbers b is comparatively easy, and series representations of ln(b) are found by coupling 2 to b with logarithmic conversions.

If ps = qt + d with some small d, then ⁠ps/qt⁠ = 1 + ⁠d/qt⁠ and therefore Selecting q = 2 represents ln p by ln 2 and a series of a parameter ⁠d/qt⁠ that one wishes to keep small for quick convergence.

Taking 32 = 23 + 1, for example, generates This is actually the third line in the following table of expansions of this type: Starting from the natural logarithm of q = 10 one might use these parameters: This is a table of recent records in calculating digits of ln 2.