Natural logarithm of 2
The decimal value of the natural logarithm of 2
is approximately
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 Lindemann–Weierstrass theorem, the natural logarithm of any natural number other than 0 and 1 is a transcendental number.
Series representations
Rising alternate factorial
Binary rising constant factorial
Other series representations
Involving the Riemann Zeta function
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:
Representation as integrals
The natural logarithm of 2 occurs frequently as the result of integration. Some explicit formulas for it include:Other representations
The Pierce expansion isThe 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:
Bootstrapping other logarithms
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 factorizationsThis employs
| prime | approximate natural logarithm | OEIS |
| 2 | ||
| 3 | ||
| 5 | ||
| 7 | ||
| 11 | ||
| 13 | ||
| 17 | ||
| 19 | ||
| 23 | ||
| 29 | ||
| 31 | ||
| 37 | ||
| 41 | ||
| 43 | ||
| 47 | ||
| 53 | ||
| 59 | ||
| 61 | ||
| 67 | ||
| 71 | ||
| 73 | ||
| 79 | ||
| 83 | ||
| 89 | ||
| 97 |
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.
Example
If with some small, then and thereforeSelecting 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:
| 1 | 3 | 1 | 2 | = … |
| 1 | 3 | 2 | 2 | − = −… |
| 2 | 3 | 3 | 2 | = … |
| 5 | 3 | 8 | 2 | − = −… |
| 12 | 3 | 19 | 2 | = … |
| 1 | 5 | 2 | 2 | = … |
| 3 | 5 | 7 | 2 | − = −… |
| 1 | 7 | 2 | 2 | = … |
| 1 | 7 | 3 | 2 | − = −… |
| 5 | 7 | 14 | 2 | = … |
| 1 | 11 | 3 | 2 | = … |
| 2 | 11 | 7 | 2 | − = −… |
| 11 | 11 | 38 | 2 | = … |
| 1 | 13 | 3 | 2 | = … |
| 1 | 13 | 4 | 2 | − = −… |
| 3 | 13 | 11 | 2 | = … |
| 7 | 13 | 26 | 2 | − = −… |
| 10 | 13 | 37 | 2 | = … |
| 1 | 17 | 4 | 2 | = … |
| 1 | 19 | 4 | 2 | = … |
| 4 | 19 | 17 | 2 | − = −… |
| 1 | 23 | 4 | 2 | = … |
| 1 | 23 | 5 | 2 | − = −… |
| 2 | 23 | 9 | 2 | = … |
| 1 | 29 | 4 | 2 | = … |
| 1 | 29 | 5 | 2 | − = −… |
| 7 | 29 | 34 | 2 | = … |
| 1 | 31 | 5 | 2 | − = −… |
| 1 | 37 | 5 | 2 | = … |
| 4 | 37 | 21 | 2 | − = −… |
| 5 | 37 | 26 | 2 | = … |
| 1 | 41 | 5 | 2 | = … |
| 2 | 41 | 11 | 2 | − = −… |
| 3 | 41 | 16 | 2 | = … |
| 1 | 43 | 5 | 2 | = … |
| 2 | 43 | 11 | 2 | − = −… |
| 5 | 43 | 27 | 2 | = … |
| 7 | 43 | 38 | 2 | − = −… |
Starting from the natural logarithm of one might use these parameters:
| 10 | 2 | 3 | 10 | = … |
| 21 | 3 | 10 | 10 | = … |
| 3 | 5 | 2 | 10 | = … |
| 10 | 5 | 7 | 10 | − = −… |
| 6 | 7 | 5 | 10 | = … |
| 13 | 7 | 11 | 10 | − = −… |
| 1 | 11 | 1 | 10 | = … |
| 1 | 13 | 1 | 10 | = … |
| 8 | 13 | 9 | 10 | − = −… |
| 9 | 13 | 10 | 10 | = … |
| 1 | 17 | 1 | 10 | = … |
| 4 | 17 | 5 | 10 | − = −… |
| 9 | 17 | 11 | 10 | = … |
| 3 | 19 | 4 | 10 | − = −… |
| 4 | 19 | 5 | 10 | = … |
| 7 | 19 | 9 | 10 | − = −… |
| 2 | 23 | 3 | 10 | − = −… |
| 3 | 23 | 4 | 10 | = … |
| 2 | 29 | 3 | 10 | − = −… |
| 2 | 31 | 3 | 10 | − = −… |
Known digits
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 logarithmof a natural number, except that of 1.