Natural logarithm of 2
The decimal value of the natural logarithm of 2 (sequence A002162 in the OEIS) is approximately
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.
Series representations
Rising alternate factorial
- This is the well-known "alternating harmonic series".
Binary rising constant factorial
Other series representations
- using
- (sums of the reciprocals of decagonal numbers)
Involving the Riemann Zeta function
(γ is the Euler–Mascheroni constant and ζ Riemann's zeta function.)
BBP-type representations
(See more about Bailey–Borwein–Plouffe (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 is OEIS: A091846
The Engel expansion is OEIS: A059180
The cotangent expansion is OEIS: A081785
The simple continued fraction expansion is OEIS: A016730
- ,
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:
- ,[1]
- also expressible as
Bootstrapping other logarithms
Given a value of ln 2, 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 c based on their factorizations
This employs
prime | approximate natural logarithm | OEIS |
---|---|---|
2 | 0.693147180559945309417232121458 | A002162 |
3 | 1.09861228866810969139524523692 | A002391 |
5 | 1.60943791243410037460075933323 | A016628 |
7 | 1.94591014905531330510535274344 | A016630 |
11 | 2.39789527279837054406194357797 | A016634 |
13 | 2.56494935746153673605348744157 | A016636 |
17 | 2.83321334405621608024953461787 | A016640 |
19 | 2.94443897916644046000902743189 | A016642 |
23 | 3.13549421592914969080675283181 | A016646 |
29 | 3.36729582998647402718327203236 | A016652 |
31 | 3.43398720448514624592916432454 | A016654 |
37 | 3.61091791264422444436809567103 | A016660 |
41 | 3.71357206670430780386676337304 | A016664 |
43 | 3.76120011569356242347284251335 | A016666 |
47 | 3.85014760171005858682095066977 | A016670 |
53 | 3.97029191355212183414446913903 | A016676 |
59 | 4.07753744390571945061605037372 | A016682 |
61 | 4.11087386417331124875138910343 | A016684 |
67 | 4.20469261939096605967007199636 | A016690 |
71 | 4.26267987704131542132945453251 | A016694 |
73 | 4.29045944114839112909210885744 | A016696 |
79 | 4.36944785246702149417294554148 | A016702 |
83 | 4.41884060779659792347547222329 | A016706 |
89 | 4.48863636973213983831781554067 | A016712 |
97 | 4.57471097850338282211672162170 | A016720 |
In a third layer, the logarithms of rational numbers r = a/b are computed with ln(r) = ln(a) − ln(b), and logarithms of roots via ln n√c = 1/n ln(c).
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.
Example
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:
s | p | t | q | d/qt |
---|---|---|---|---|
1 | 3 | 1 | 2 | 1/2 = 0.50000000… |
1 | 3 | 2 | 2 | −1/4 = −0.25000000… |
2 | 3 | 3 | 2 | 1/8 = 0.12500000… |
5 | 3 | 8 | 2 | −13/256 = −0.05078125… |
12 | 3 | 19 | 2 | 7153/524288 = 0.01364326… |
1 | 5 | 2 | 2 | 1/4 = 0.25000000… |
3 | 5 | 7 | 2 | −3/128 = −0.02343750… |
1 | 7 | 2 | 2 | 3/4 = 0.75000000… |
1 | 7 | 3 | 2 | −1/8 = −0.12500000… |
5 | 7 | 14 | 2 | 423/16384 = 0.02581787… |
1 | 11 | 3 | 2 | 3/8 = 0.37500000… |
2 | 11 | 7 | 2 | −7/128 = −0.05468750… |
11 | 11 | 38 | 2 | 10433763667/274877906944 = 0.03795781… |
1 | 13 | 3 | 2 | 5/8 = 0.62500000… |
1 | 13 | 4 | 2 | −3/16 = −0.18750000… |
3 | 13 | 11 | 2 | 149/2048 = 0.07275391… |
7 | 13 | 26 | 2 | −4360347/67108864 = −0.06497423… |
10 | 13 | 37 | 2 | 419538377/137438953472 = 0.00305254… |
1 | 17 | 4 | 2 | 1/16 = 0.06250000… |
1 | 19 | 4 | 2 | 3/16 = 0.18750000… |
4 | 19 | 17 | 2 | −751/131072 = −0.00572968… |
1 | 23 | 4 | 2 | 7/16 = 0.43750000… |
1 | 23 | 5 | 2 | −9/32 = −0.28125000… |
2 | 23 | 9 | 2 | 17/512 = 0.03320312… |
1 | 29 | 4 | 2 | 13/16 = 0.81250000… |
1 | 29 | 5 | 2 | −3/32 = −0.09375000… |
7 | 29 | 34 | 2 | 70007125/17179869184 = 0.00407495… |
1 | 31 | 5 | 2 | −1/32 = −0.03125000… |
1 | 37 | 5 | 2 | 5/32 = 0.15625000… |
4 | 37 | 21 | 2 | −222991/2097152 = −0.10633039… |
5 | 37 | 26 | 2 | 2235093/67108864 = 0.03330548… |
1 | 41 | 5 | 2 | 9/32 = 0.28125000… |
2 | 41 | 11 | 2 | −367/2048 = −0.17919922… |
3 | 41 | 16 | 2 | 3385/65536 = 0.05165100… |
1 | 43 | 5 | 2 | 11/32 = 0.34375000… |
2 | 43 | 11 | 2 | −199/2048 = −0.09716797… |
5 | 43 | 27 | 2 | 12790715/134217728 = 0.09529825… |
7 | 43 | 38 | 2 | −3059295837/274877906944 = −0.01112965… |
Starting from the natural logarithm of q = 10 one might use these parameters:
s | p | t | q | d/qt |
---|---|---|---|---|
10 | 2 | 3 | 10 | 3/125 = 0.02400000… |
21 | 3 | 10 | 10 | 460353203/10000000000 = 0.04603532… |
3 | 5 | 2 | 10 | 1/4 = 0.25000000… |
10 | 5 | 7 | 10 | −3/128 = −0.02343750… |
6 | 7 | 5 | 10 | 17649/100000 = 0.17649000… |
13 | 7 | 11 | 10 | −3110989593/100000000000 = −0.03110990… |
1 | 11 | 1 | 10 | 1/10 = 0.10000000… |
1 | 13 | 1 | 10 | 3/10 = 0.30000000… |
8 | 13 | 9 | 10 | −184269279/1000000000 = −0.18426928… |
9 | 13 | 10 | 10 | 604499373/10000000000 = 0.06044994… |
1 | 17 | 1 | 10 | 7/10 = 0.70000000… |
4 | 17 | 5 | 10 | −16479/100000 = −0.16479000… |
9 | 17 | 11 | 10 | 18587876497/100000000000 = 0.18587876… |
3 | 19 | 4 | 10 | −3141/10000 = −0.31410000… |
4 | 19 | 5 | 10 | 30321/100000 = 0.30321000… |
7 | 19 | 9 | 10 | −106128261/1000000000 = −0.10612826… |
2 | 23 | 3 | 10 | −471/1000 = −0.47100000… |
3 | 23 | 4 | 10 | 2167/10000 = 0.21670000… |
2 | 29 | 3 | 10 | −159/1000 = −0.15900000… |
2 | 31 | 3 | 10 | −39/1000 = −0.03900000… |
Known digits
This is a table of recent records in calculating digits of ln 2. As of December 2018, it has been calculated to more digits than any other natural logarithm [2] [3] of a natural number, except that of 1.
Date | Name | Number of digits |
---|---|---|
January 7, 2009 | A.Yee & R.Chan | 15,500,000,000 |
February 4, 2009 | A.Yee & R.Chan | 31,026,000,000 |
February 21, 2011 | Alexander Yee | 50,000,000,050 |
May 14, 2011 | Shigeru Kondo | 100,000,000,000 |
February 28, 2014 | Shigeru Kondo | 200,000,000,050 |
July 12, 2015 | Ron Watkins | 250,000,000,000 |
January 30, 2016 | Ron Watkins | 350,000,000,000 |
April 18, 2016 | Ron Watkins | 500,000,000,000 |
December 10, 2018 | Michael Kwok | 600,000,000,000 |
April 26, 2019 | Jacob Riffee | 1,000,000,000,000 |
August 19, 2020 | Seungmin Kim[4][5] | 1,200,000,000,100 |
See also
- Rule of 72#Continuous compounding, in which ln 2 figures prominently
- Half-life#Formulas for half-life in exponential decay, in which ln 2 figures prominently
- Erdős–Moser equation: all solutions must come from a convergent of ln 2.
References
- Brent, Richard P. (1976). "Fast multiple-precision evaluation of elementary functions". J. ACM. 23 (2): 242–251. doi:10.1145/321941.321944. MR 0395314.
- Uhler, Horace S. (1940). "Recalculation and extension of the modulus and of the logarithms of 2, 3, 5, 7 and 17". Proc. Natl. Acad. Sci. U.S.A. 26 (3): 205–212. doi:10.1073/pnas.26.3.205. MR 0001523. PMC 1078033. PMID 16588339.
- Sweeney, Dura W. (1963). "On the computation of Euler's constant". Mathematics of Computation. 17 (82): 170–178. doi:10.1090/S0025-5718-1963-0160308-X. MR 0160308.
- Chamberland, Marc (2003). "Binary BBP-formulae for logarithms and generalized Gaussian–Mersenne primes" (PDF). Journal of Integer Sequences. 6: 03.3.7. MR 2046407. Archived from the original (PDF) on 2011-06-06. Retrieved 2010-04-29.
- Gourévitch, Boris; Guillera Goyanes, Jesús (2007). "Construction of binomial sums for π and polylogarithmic constants inspired by BBP formulas" (PDF). Applied Math. E-Notes. 7: 237–246. MR 2346048.
- Wu, Qiang (2003). "On the linear independence measure of logarithms of rational numbers". Mathematics of Computation. 72 (242): 901–911. doi:10.1090/S0025-5718-02-01442-4.
- Borwein, J.; Crandall, R.; Free, G. (2004). "On the Ramanujan AGM Fraction , I: The Real-Parameter Case" (PDF). Exper. Math. 13 (3): 278–280. doi:10.1080/10586458.2004.10504540.
- "y-cruncher". numberworld.org. Retrieved 10 December 2018.
- "Natural log of 2". numberworld.org. Retrieved 10 December 2018.
- "Records set by y-cruncher". Archived from the original on 2020-09-15. Retrieved September 15, 2020.
- "Natural logarithm of 2 (Log(2)) world record by Seungmin Kim". Retrieved September 15, 2020.
External links
- Weisstein, Eric W. "Natural logarithm of 2". MathWorld.
- "table of natural logarithms". PlanetMath.
- Gourdon, Xavier; Sebah, Pascal. "The logarithm constant:log 2".