The decimal value of the natural logarithm of 2 (sequence A002162 OEIS ) is approximately
ln
(
2
)
≈
0.693
147
180
56
{\displaystyle \ln(2)\approx 0.693\,147\,180\,56}
as shown in the first line of the table below. The logarithm in other bases is obtained with the formula
log
b
(
2
)
=
ln
(
2
)
ln
(
b
)
.
{\displaystyle \log _{b}(2)={\frac {\ln(2)}{\ln(b)}}.}
The common logarithm in particular is (A007524
log
10
(
2
)
≈
0.301
029
995
663
981
195.
{\displaystyle \log _{10}(2)\approx 0.301\,029\,995\,663\,981\,195.}
The inverse of this number is the binary logarithm of 10:
log
2
(
10
)
=
1
log
10
(
2
)
≈
3.321
928
095
{\displaystyle \log _{2}(10)={\frac {1}{\log _{10}(2)}}\approx 3.321\,928\,095}
A020862
number
approximate natural logarithm
OEIS
2 6999693147180559945♠ 0.693147 180 559 945 309 417 232 121 458 A002162
3 7000109861228866810♠ 1.098612 288 668 109 691 395 245 236 92 A002391
4 7000138629436111989♠ 1.386294 361 119 890 618 834 464 242 92 A016627
5 7000160943791243410♠ 1.609437 912 434 100 374 600 759 333 23 A016628
6 7000179175946922805♠ 1.791759 469 228 055 000 812 477 358 38 A016629
7 7000194591014905531♠ 1.945910 149 055 313 305 105 352 743 44 A016630
8 7000207944154167983♠ 2.079441 541 679 835 928 251 696 364 37 A016631
9 7000219722457733621♠ 2.197224 577 336 219 382 790 490 473 84 A016632
10 7000230258509299404♠ 2.302585 092 994 045 684 017 991 454 68 A002392
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 [ edit ]
∑
n
=
1
∞
(
−
1
)
n
+
1
n
=
∑
n
=
0
∞
1
(
2
n
+
1
)
(
2
n
+
2
)
=
ln
2.
{\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n+1}}{n}}=\sum _{n=0}^{\infty }{\frac {1}{(2n+1)(2n+2)}}=\ln 2.}
∑
n
=
0
∞
(
−
1
)
n
(
n
+
1
)
(
n
+
2
)
=
2
ln
(
2
)
−
1.
{\displaystyle \sum _{n=0}^{\infty }{\frac {(-1)^{n}}{(n+1)(n+2)}}=2\ln(2)-1.}
∑
n
=
1
∞
1
n
(
4
n
2
−
1
)
=
2
ln
(
2
)
−
1.
{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{n(4n^{2}-1)}}=2\ln(2)-1.}
∑
n
=
1
∞
(
−
1
)
n
n
(
4
n
2
−
1
)
=
ln
(
2
)
−
1.
{\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n}}{n(4n^{2}-1)}}=\ln(2)-1.}
∑
n
=
1
∞
(
−
1
)
n
n
(
9
n
2
−
1
)
=
2
ln
(
2
)
−
3
2
.
{\displaystyle \sum _{n=1}^{\infty }{\frac {(-1)^{n}}{n(9n^{2}-1)}}=2\ln(2)-{\tfrac {3}{2}}.}
∑
n
=
2
∞
1
2
n
[
ζ
(
n
)
−
1
]
=
ln
(
2
)
−
1
2
.
{\displaystyle \sum _{n=2}^{\infty }{\frac {1}{2^{n}}}[\zeta (n)-1]=\ln(2)-{\tfrac {1}{2}}.}
∑
n
=
1
∞
1
2
n
+
1
[
ζ
(
n
)
−
1
]
=
1
−
γ
−
ln
(
2
)
2
.
{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{2n+1}}[\zeta (n)-1]=1-\gamma -{\frac {\ln(2)}{2}}.}
∑
n
=
1
∞
1
2
2
n
(
2
n
+
1
)
ζ
(
2
n
)
=
1
−
ln
(
2
)
2
.
{\displaystyle \sum _{n=1}^{\infty }{\frac {1}{2^{2n}(2n+1)}}\zeta (2n)={\frac {1-\ln(2)}{2}}.}
ln
(
2
)
=
∑
n
=
1
∞
1
2
n
n
.
{\displaystyle \ln(2)=\sum _{n=1}^{\infty }{\frac {1}{2^{n}n}}.}
ln
(
2
)
=
∑
n
=
1
∞
(
1
3
n
+
1
4
n
)
1
n
.
{\displaystyle \ln(2)=\sum _{n=1}^{\infty }\left({\frac {1}{3^{n}}}+{\frac {1}{4^{n}}}\right){\frac {1}{n}}.}
ln
(
2
)
=
2
3
+
1
2
∑
k
≥
1
(
1
2
k
+
1
4
k
+
1
+
1
8
k
+
4
+
1
16
k
+
12
)
1
16
k
.
{\displaystyle \ln(2)={\frac {2}{3}}+{\frac {1}{2}}\sum _{k\geq 1}\left({\frac {1}{2k}}+{\frac {1}{4k+1}}+{\frac {1}{8k+4}}+{\frac {1}{16k+12}}\right){\frac {1}{16^{k}}}.}
ln
(
2
)
=
2
3
∑
k
≥
0
1
9
k
(
2
k
+
1
)
.
{\displaystyle \ln(2)={\frac {2}{3}}\sum _{k\geq 0}{\frac {1}{9^{k}(2k+1)}}.}
ln
(
2
)
=
∑
k
≥
0
(
14
31
2
k
+
1
(
2
k
+
1
)
+
6
161
2
k
+
1
(
2
k
+
1
)
+
10
49
2
k
+
1
(
2
k
+
1
)
)
.
{\displaystyle \ln(2)=\sum _{k\geq 0}\left({\frac {14}{31^{2k+1}(2k+1)}}+{\frac {6}{161^{2k+1}(2k+1)}}+{\frac {10}{49^{2k+1}(2k+1)}}\right).}
ln
(
2
)
=
∑
n
=
1
∞
1
4
n
2
−
2
n
{\displaystyle \ln(2)=\sum _{n=1}^{\infty }{\frac {1}{4n^{2}-2n}}}
ln
(
2
)
=
∑
n
=
1
∞
2
(
−
1
)
n
+
1
(
2
n
−
1
)
+
1
8
n
2
−
4
n
{\displaystyle \ln(2)=\sum _{n=1}^{\infty }{\frac {2(-1)^{n+1}(2n-1)+1}{8n^{2}-4n}}}
(γ Euler–Mascheroni constant and ζ Riemann's zeta function .)
Some Bailey–Borwein–Plouffe (BBP)-type representations fall also into this category.
Representation as integrals [ edit ]
∫
0
1
d
x
1
+
x
=
ln
(
2
)
,
or, equivalently,
∫
1
2
d
x
x
=
ln
(
2
)
.
{\displaystyle \int _{0}^{1}{\frac {dx}{1+x}}=\ln(2),{\text{ or, equivalently, }}\int _{1}^{2}{\frac {dx}{x}}=\ln(2).}
∫
1
∞
d
x
(
1
+
x
2
)
(
1
+
x
)
2
=
1
−
ln
(
2
)
4
.
{\displaystyle \int _{1}^{\infty }{\frac {dx}{(1+x^{2})(1+x)^{2}}}={\frac {1-\ln(2)}{4}}.}
∫
0
∞
d
x
1
+
e
n
x
=
ln
(
2
)
n
;
∫
0
∞
d
x
3
+
e
n
x
=
2
ln
(
2
)
3
n
.
{\displaystyle \int _{0}^{\infty }{\frac {dx}{1+e^{nx}}}={\frac {\ln(2)}{n}};\int _{0}^{\infty }{\frac {dx}{3+e^{nx}}}={\frac {2\ln(2)}{3n}}.}
∫
0
∞
1
e
x
−
1
−
2
e
2
x
−
1
d
x
=
ln
(
2
)
.
{\displaystyle \int _{0}^{\infty }{\frac {1}{e^{x}-1}}-{\frac {2}{e^{2x}-1}}\,dx=\ln(2).}
∫
0
∞
e
−
x
1
−
e
−
x
x
d
x
=
ln
(
2
)
.
{\displaystyle \int _{0}^{\infty }e^{-x}{\frac {1-e^{-x}}{x}}\,dx=\ln(2).}
∫
0
1
ln
(
x
2
−
1
x
ln
(
x
)
)
d
x
=
−
1
+
ln
(
2
)
+
γ
.
{\displaystyle \int _{0}^{1}\ln \left({\frac {x^{2}-1}{x\ln(x)}}\right)dx=-1+\ln(2)+\gamma .}
∫
0
π
3
tan
(
x
)
d
x
=
2
∫
0
π
4
tan
(
x
)
d
x
=
ln
(
2
)
.
{\displaystyle \int _{0}^{\frac {\pi }{3}}\tan(x)\,dx=2\int _{0}^{\frac {\pi }{4}}\tan(x)\,dx=\ln(2).}
∫
−
π
4
π
4
ln
(
sin
(
x
)
+
cos
(
x
)
)
d
x
=
−
π
ln
(
2
)
4
.
{\displaystyle \int _{-{\frac {\pi }{4}}}^{\frac {\pi }{4}}\ln \left(\sin(x)+\cos(x)\right)\,dx=-{\frac {\pi \ln(2)}{4}}.}
∫
0
1
x
2
ln
(
1
+
x
)
d
x
=
2
ln
(
2
)
3
−
5
18
.
{\displaystyle \int _{0}^{1}x^{2}\ln(1+x)\,dx={\frac {2\ln(2)}{3}}-{\frac {5}{18}}.}
∫
0
1
x
ln
(
1
+
x
)
ln
(
1
−
x
)
d
x
=
1
4
−
ln
(
2
)
.
{\displaystyle \int _{0}^{1}x\ln(1+x)\ln(1-x)\,dx={\tfrac {1}{4}}-\ln(2).}
∫
0
1
x
3
ln
(
1
+
x
)
ln
(
1
−
x
)
d
x
=
13
96
−
2
ln
(
2
)
3
.
{\displaystyle \int _{0}^{1}x^{3}\ln(1+x)\ln(1-x)\,dx={\tfrac {13}{96}}-{\frac {2\ln(2)}{3}}.}
∫
0
1
ln
x
(
1
+
x
)
2
d
x
=
−
ln
(
2
)
.
{\displaystyle \int _{0}^{1}{\frac {\ln x}{(1+x)^{2}}}\,dx=-\ln(2).}
∫
0
1
ln
(
1
+
x
)
−
x
x
2
d
x
=
1
−
2
ln
(
2
)
.
{\displaystyle \int _{0}^{1}{\frac {\ln(1+x)-x}{x^{2}}}\,dx=1-2\ln(2).}
∫
0
1
d
x
x
(
1
−
ln
(
x
)
)
(
1
−
2
ln
(
x
)
)
=
ln
(
2
)
.
{\displaystyle \int _{0}^{1}{\frac {dx}{x(1-\ln(x))(1-2\ln(x))}}=\ln(2).}
∫
1
∞
ln
(
ln
(
x
)
)
x
3
d
x
=
−
γ
+
ln
(
2
)
2
.
{\displaystyle \int _{1}^{\infty }{\frac {\ln \left(\ln(x)\right)}{x^{3}}}\,dx=-{\frac {\gamma +\ln(2)}{2}}.}
(γ Euler–Mascheroni constant .)
Other representations [ edit ] The Pierce expansion is A091846
ln
(
2
)
=
1
−
1
1
⋅
3
+
1
1
⋅
3
⋅
12
−
⋯
.
{\displaystyle \ln(2)=1-{\frac {1}{1\cdot 3}}+{\frac {1}{1\cdot 3\cdot 12}}-\cdots .}
The Engel expansion is A059180
ln
(
2
)
=
1
2
+
1
2
⋅
3
+
1
2
⋅
3
⋅
7
+
1
2
⋅
3
⋅
7
⋅
9
+
⋯
.
{\displaystyle \ln(2)={\frac {1}{2}}+{\frac {1}{2\cdot 3}}+{\frac {1}{2\cdot 3\cdot 7}}+{\frac {1}{2\cdot 3\cdot 7\cdot 9}}+\cdots .}
The cotangent expansion is A081785
ln
(
2
)
=
cot
(
arccot
(
0
)
−
arccot
(
1
)
+
arccot
(
5
)
−
arccot
(
55
)
+
arccot
(
14187
)
−
⋯
)
.
{\displaystyle \ln(2)=\cot({\operatorname {arccot}(0)-\operatorname {arccot}(1)+\operatorname {arccot}(5)-\operatorname {arccot}(55)+\operatorname {arccot}(14187)-\cdots }).}
As an infinite sum of fractions:[1]
ln
(
2
)
=
1
−
1
2
+
1
3
−
1
4
+
1
5
−
⋯
.
{\displaystyle \ln(2)=1-{\tfrac {1}{2}}+{\tfrac {1}{3}}-{\tfrac {1}{4}}+{\tfrac {1}{5}}-\cdots .}
It can also be expressed through the Taylor series :
ln
(
2
)
=
1
2
+
1
12
+
1
30
+
1
56
+
1
90
+
⋯
{\textstyle \quad \ln(2)={\tfrac {1}{2}}+{\tfrac {1}{12}}+{\tfrac {1}{30}}+{\tfrac {1}{56}}+{\tfrac {1}{90}}+\cdots }
This generalized continued fraction :
ln
(
2
)
=
[
0
;
1
,
2
,
3
,
1
,
5
,
2
3
,
7
,
1
2
,
9
,
2
5
,
.
.
.
,
2
k
−
1
,
2
k
,
.
.
.
]
{\displaystyle \ln(2)=\left[0;1,2,3,1,5,{\tfrac {2}{3}},7,{\tfrac {1}{2}},9,{\tfrac {2}{5}},...,2k-1,{\frac {2}{k}},...\right]}
[2] also expressible as
ln
(
2
)
=
1
1
+
1
2
+
1
3
+
2
2
+
2
5
+
3
2
+
3
7
+
4
2
+
⋱
=
2
3
−
1
2
9
−
2
2
15
−
3
2
21
−
⋱
{\displaystyle \ln(2)={\cfrac {1}{1+{\cfrac {1}{2+{\cfrac {1}{3+{\cfrac {2}{2+{\cfrac {2}{5+{\cfrac {3}{2+{\cfrac {3}{7+{\cfrac {4}{2+\ddots }}}}}}}}}}}}}}}}={\cfrac {2}{3-{\cfrac {1^{2}}{9-{\cfrac {2^{2}}{15-{\cfrac {3^{2}}{21-\ddots }}}}}}}}}
Bootstrapping other logarithms [ edit ] 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 factorizations
c
=
2
i
3
j
5
k
7
l
⋯
→
ln
(
c
)
=
i
ln
(
2
)
+
j
ln
(
3
)
+
k
ln
(
5
)
+
l
ln
(
7
)
+
⋯
{\displaystyle c=2^{i}3^{j}5^{k}7^{l}\cdots \rightarrow \ln(c)=i\ln(2)+j\ln(3)+k\ln(5)+l\ln(7)+\cdots }
Apart from the logarithms of 2, 3, 5 and 7 shown above, this employs
prime
approximate natural logarithm
OEIS
11 7000239789527279837♠ 2.397895 272 798 370 544 061 943 577 97 A016634
13 7000256494935746153♠ 2.564949 357 461 536 736 053 487 441 57 A016636
17 7000283321334405621♠ 2.833213 344 056 216 080 249 534 617 87 A016640
19 7000294443897916644♠ 2.944438 979 166 440 460 009 027 431 89 A016642
23 7000313549421592914♠ 3.135494 215 929 149 690 806 752 831 81 A016646
29 7000336729582998647♠ 3.367295 829 986 474 027 183 272 032 36 A016652
31 7000343398720448514♠ 3.433987 204 485 146 245 929 164 324 54 A016654
37 7000361091791264422♠ 3.610917 912 644 224 444 368 095 671 03 A016660
41 7000371357206670430♠ 3.713572 066 704 307 803 866 763 373 04 A016664
43 7000376120011569356♠ 3.761200 115 693 562 423 472 842 513 35 A016666
47 7000385014760171005♠ 3.850147 601 710 058 586 820 950 669 77 A016670
53 7000397029191355212♠ 3.970291 913 552 121 834 144 469 139 03 A016676
59 7000407753744390572♠ 4.077537 443 905 719 450 616 050 373 72 A016682
61 7000411087386417331♠ 4.110873 864 173 311 248 751 389 103 43 A016684
67 7000420469261939096♠ 4.204692 619 390 966 059 670 071 996 36 A016690
71 7000426267987704131♠ 4.262679 877 041 315 421 329 454 532 51 A016694
73 7000429045944114839♠ 4.290459 441 148 391 129 092 108 857 44 A016696
79 7000436944785246702♠ 4.369447 852 467 021 494 172 945 541 48 A016702
83 7000441884060779659♠ 4.418840 607 796 597 923 475 472 223 29 A016706
89 7000448863636973213♠ 4.488636 369 732 139 838 317 815 540 67 A016712
97 7000457471097850338♠ 4.574710 978 503 382 822 116 721 621 70 A016720
In a third layer, the logarithms of rational numbers r = a / b ln(r ) = ln(a ) − ln(b ) , and logarithms of roots via ln n c 1 / n 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 b ln(b ) are found by coupling 2 to b logarithmic conversions .
Example [ edit ] If ps = qt + d d ps / qt d / qt
s
ln
(
p
)
−
t
ln
(
q
)
=
ln
(
1
+
d
q
t
)
=
∑
m
=
1
∞
(
−
1
)
m
+
1
(
d
q
t
)
m
m
.
{\displaystyle s\ln(p)-t\ln(q)=\ln \left(1+{\frac {d}{q^{t}}}\right)=\sum _{m=1}^{\infty }(-1)^{m+1}{\frac {({\frac {d}{q^{t}}})^{m}}{m}}.}
Selecting q = 2ln(p ) by ln(2) and a series of a parameter d / qt 32 = 23 + 1 , for example, generates
2
ln
(
3
)
=
3
ln
(
2
)
−
∑
k
≥
1
(
−
1
)
k
8
k
k
.
{\displaystyle 2\ln(3)=3\ln(2)-\sum _{k\geq 1}{\frac {(-1)^{k}}{8^{k}k}}.}
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 − 6999500000000000000♠ 0.500000 00
1
3
2
2
−1 / 4 6999250000000000000♠ 0.250000 00
2
3
3
2
1 / 8 − 6999125000000000000♠ 0.125000 00
5
3
8
2
−13 / 256 6998507812500000000♠ 0.050781 25
12
3
19
2
7153 / 7005524288000000000♠ 524288 − 6998136432600000000♠ 0.013643 26
1
5
2
2
1 / 4 − 6999250000000000000♠ 0.250000 00
3
5
7
2
−3 / 128 6998234375000000000♠ 0.023437 50
1
7
2
2
3 / 4 − 6999750000000000000♠ 0.750000 00
1
7
3
2
−1 / 8 6999125000000000000♠ 0.125000 00
5
7
14
2
423 / 7004163840000000000♠ 16384 − 6998258178700000000♠ 0.025817 87
1
11
3
2
3 / 8 − 6999375000000000000♠ 0.375000 00
2
11
7
2
−7 / 128 6998546875000000000♠ 0.054687 50
11
11
38
2
7010104337636670000♠ 10433 763 667 / 7011274877906944000♠ 274877 906 944 − 6998379578100000000♠ 0.037957 81
1
13
3
2
5 / 8 − 6999625000000000000♠ 0.625000 00
1
13
4
2
−3 / 16 6999187500000000000♠ 0.187500 00
3
13
11
2
149 / 2048 − 6998727539100000000♠ 0.072753 91
7
13
26
2
−7006436034700000000♠ 4360 347 / 7007671088640000000♠ 67108 864 6998649742300000000♠ 0.064974 23
10
13
37
2
7008419538377000000♠ 419538 377 / 7011137438953472000♠ 137438 953 472 − 6997305254000000000♠ 0.003052 54
1
17
4
2
1 / 16 − 6998625000000000000♠ 0.062500 00
1
19
4
2
3 / 16 − 6999187500000000000♠ 0.187500 00
4
19
17
2
−751 / 7005131072000000000♠ 131072 6997572968000000000♠ 0.005729 68
1
23
4
2
7 / 16 − 6999437500000000000♠ 0.437500 00
1
23
5
2
−9 / 32 6999281250000000000♠ 0.281250 00
2
23
9
2
17 / 512 − 6998332031200000000♠ 0.033203 12
1
29
4
2
13 / 16 − 6999812500000000000♠ 0.812500 00
1
29
5
2
−3 / 32 6998937500000000000♠ 0.093750 00
7
29
34
2
7007700071250000000♠ 70007 125 / 7010171798691840000♠ 17179 869 184 − 6997407495000000000♠ 0.004074 95
1
31
5
2
−1 / 32 6998312500000000000♠ 0.031250 00
1
37
5
2
5 / 32 − 6999156250000000000♠ 0.156250 00
4
37
21
2
−7005222991000000000♠ 222991 / 7006209715200000000♠ 2097 152 6999106330390000000♠ 0.106330 39
5
37
26
2
7006223509300000000♠ 2235 093 / 7007671088640000000♠ 67108 864 − 6998333054800000000♠ 0.033305 48
1
41
5
2
9 / 32 − 6999281250000000000♠ 0.281250 00
2
41
11
2
−367 / 2048 6999179199220000000♠ 0.179199 22
3
41
16
2
3385 / 7004655360000000000♠ 65536 − 6998516510000000000♠ 0.051651 00
1
43
5
2
11 / 32 − 6999343750000000000♠ 0.343750 00
2
43
11
2
−199 / 2048 6998971679700000000♠ 0.097167 97
5
43
27
2
7007127907150000000♠ 12790 715 / 7008134217728000000♠ 134217 728 − 6998952982500000000♠ 0.095298 25
7
43
38
2
−7009305929583700000♠ 3059 295 837 / 7011274877906944000♠ 274877 906 944 6998111296500000000♠ 0.011129 65
Starting from the natural logarithm of q = 10
s p t q d / qt
10
2
3
10
3 / 125 − 6998240000000000000♠ 0.024000 00
21
3
10
10
7008460353203000000♠ 460353 203 / 7010100000000000000♠ 10000 000 000 − 6998460353200000000♠ 0.046035 32
3
5
2
10
1 / 4 − 6999250000000000000♠ 0.250000 00
10
5
7
10
−3 / 128 6998234375000000000♠ 0.023437 50
6
7
5
10
7004176490000000000♠ 17649 / 7005100000000000000♠ 100000 − 6999176490000000000♠ 0.176490 00
13
7
11
10
−7009311098959300000♠ 3110 989 593 / 7011100000000000000♠ 100000 000 000 6998311099000000000♠ 0.031109 90
1
11
1
10
1 / 10 − 6999100000000000000♠ 0.100000 00
1
13
1
10
3 / 10 − 6999300000000000000♠ 0.300000 00
8
13
9
10
−7008184269279000000♠ 184269 279 / 7009100000000000000♠ 1000 000 000 6999184269280000000♠ 0.184269 28
9
13
10
10
7008604499373000000♠ 604499 373 / 7010100000000000000♠ 10000 000 000 − 6998604499400000000♠ 0.060449 94
1
17
1
10
7 / 10 − 6999700000000000000♠ 0.700000 00
4
17
5
10
−7004164790000000000♠ 16479 / 7005100000000000000♠ 100000 6999164790000000000♠ 0.164790 00
9
17
11
10
7010185878764970000♠ 18587 876 497 / 7011100000000000000♠ 100000 000 000 − 6999185878760000000♠ 0.185878 76
3
19
4
10
−3141 / 7004100000000000000♠ 10000 6999314100000000000♠ 0.314100 00
4
19
5
10
7004303210000000000♠ 30321 / 7005100000000000000♠ 100000 − 6999303210000000000♠ 0.303210 00
7
19
9
10
−7008106128261000000♠ 106128 261 / 7009100000000000000♠ 1000 000 000 6999106128260000000♠ 0.106128 26
2
23
3
10
−471 / 1000 6999471000000000000♠ 0.471000 00
3
23
4
10
2167 / 7004100000000000000♠ 10000 − 6999216700000000000♠ 0.216700 00
2
29
3
10
−159 / 1000 6999159000000000000♠ 0.159000 00
2
31
3
10
−39 / 1000 6998390000000000000♠ 0.039000 00
References [ edit ]
Brent, Richard P. (1976). "Fast multiple-precision evaluation of elementary functions". J. ACM . 23 (2): 242–251. MR 0395314 . doi :10.1145/321941.321944 . Uhler, Horace S. (1940). "Recalculation and extension of the modulus and of the logarithms of 2, 3, 5, 7 and 17" (PDF) . Proc. Natl. Acad. Sci. U.S.A . 26 : 205–212. MR 0001523 . doi :10.1073/pnas.26.3.205 . Sweeney, Dura W. (1963). "On the computation of Euler's constant". Mathematics of Computation . 17 : 170–178. MR 0160308 . doi :10.1090/S0025-5718-1963-0160308-X . Chamberland, Marc (2003). "Binary BBP-formulae for logarithms and generalized Gaussian–Mersenne primes" (PDF) . Journal of Integer Sequences . 6 : 03.3.7. MR 2046407 . 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 .
External links [ edit ] 
A007524)
A020862).
![{\displaystyle \sum _{n=2}^{\infty }{\frac {1}{2^{n}}}[\zeta (n)-1]=\ln(2)-{\tfrac {1}{2}}.}](https://web-archive.nli.org.il/National_Library/20160521004321im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/64063479a21c1cea6c0cb01305aa4432d1abcb06)
![\sum_{n=1}^\infty \frac{1}{2n+1}[\zeta(n)-1] = 1-\gamma-\frac{\ln(2)}{2}.](https://web-archive.nli.org.il/National_Library/20160521004321im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/e23cb29f2a18bab46ddecc91263692043d8ad985)
![{\displaystyle \ln(2)=\left[0;1,2,3,1,5,{\tfrac {2}{3}},7,{\tfrac {1}{2}},9,{\tfrac {2}{5}},...,2k-1,{\frac {2}{k}},...\right]}](https://web-archive.nli.org.il/National_Library/20160521004321im_/https://wikimedia.org/api/rest_v1/media/math/render/svg/b5ea1928e6e4724abfa9bd3c862f287bf29dc8dd)
