Natural logarithm of 2
The decimal value of the natural logarithm of 2 (sequence A002162 in OEIS) is approximately
as shown in the first line of the table below. The logarithm in other bases is obtained with the formula
The common logarithm in particular is ( A007524)
The inverse of this number is the binary logarithm of 10:
- ( A020862).
number | approximate natural logarithm | OEIS |
---|---|---|
2 | 0.693147180559945309417232121458 | A002162 |
3 | 1.09861228866810969139524523692 | A002391 |
4 | 1.38629436111989061883446424292 | A016627 |
5 | 1.60943791243410037460075933323 | A016628 |
6 | 1.79175946922805500081247735838 | A016629 |
7 | 1.94591014905531330510535274344 | A016630 |
8 | 2.07944154167983592825169636437 | A016631 |
9 | 2.19722457733621938279049047384 | A016632 |
10 | 2.30258509299404568401799145468 | A002392 |
Contents
Series representations
( is the Euler–Mascheroni constant and Riemann's zeta function).
Some Bailey–Borwein–Plouffe (BBP)-type representations fall also into this category.
Representation as integrals
( is the Euler–Mascheroni constant).
Other representations
The Pierce expansion is A091846
The Engel expansion is A059180
The cotangent expansion is A081785
As an infinite sum of fractions:[1]
This generalized continued fraction:
- ,[2]
- also expressible as
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 factorizations
Apart from the logarithms of 2, 3, 5 and 7 shown above, this employs
prime | approximate natural logarithm | OEIS |
---|---|---|
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 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 to with logarithmic conversions.
Example
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:
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 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... |
References
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External links
- Weisstein, Eric W., "Natural logarithm of 2", MathWorld.
- table of natural logarithms at PlanetMath.org.
- Lua error in package.lua at line 80: module 'strict' not found.