In mathematics, the natural logarithm of 2 is the unique real number argument such that the exponential function equals two. It appears frequently in various formulas and is also given by the alternating harmonic series. The decimal value of the natural logarithm of 2 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 ()
The inverse of this number is the binary logarithm of 10:
By the LindemannâÂÂWeierstrass 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. It is also contained in the ring of algebraic periods.
( is the EulerâÂÂMascheroni constant and Riemann's zeta function.)
(See more about BaileyâÂÂBorweinâÂÂPlouffe (BBP)-type representations.)
Applying the three general series for natural logarithm to 2 directly gives:
Applying them to gives:
Applying them to gives:
Applying them to gives:
The natural logarithm of 2 occurs frequently as the result of integration. Some explicit formulas for it include:
The Pierce expansion is
The Engel expansion is
The cotangent expansion is
The simple continued fraction expansion is
which yields rational approximations, the first few of which are 0, 1, 2/3, 7/10, 9/13 and 61/88.
This generalized continued fraction:
The following continued fraction representation (J.L.Lagrange) gives (asymptotically) 1.53 new correct decimal places per cycle:
or
Given a value of , a scheme of computing the logarithms of other integers is to tabulate the logarithms of the prime numbers and in the next layer the logarithms of the composite numbers based on their factorizations
This employs
In a third layer, the logarithms of rational numbers are computed with , and logarithms of roots via .
The logarithm of 2 is useful in the sense that the powers of 2 are rather densely distributed; finding powers close to powers of other numbers is comparatively easy, and series representations of are found by coupling 2 to with logarithmic conversions.
If with some small , then and therefore
Selecting represents by and a series of a parameter that one wishes to keep small for quick convergence. Taking , for example, generates
This is actually the third line in the following table of expansions of this type:
Starting from the natural logarithm of one might use these parameters:
This is a table of recent records in calculating digits of . As of December 2018, it has been calculated to more digits than any other natural logarithm of a natural number, except that of 1.