The decimal value of the natural logarithm of 2 (sequence A002162 in the OEIS) is approximately
ln
(
2
)
≈
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
)
.
The common logarithm in particular is ( A007524)
log
10
(
2
)
≈
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
(
A020862).
By 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.
∑
n
=
1
∞
(
−
1
)
n
+
1
n
=
∑
n
=
0
∞
1
(
2
n
+
1
)
(
2
n
+
2
)
=
ln
2.
∑
n
=
0
∞
(
−
1
)
n
(
n
+
1
)
(
n
+
2
)
=
2
ln
(
2
)
−
1.
∑
n
=
1
∞
1
n
(
4
n
2
−
1
)
=
2
ln
(
2
)
−
1.
∑
n
=
1
∞
(
−
1
)
n
n
(
4
n
2
−
1
)
=
ln
(
2
)
−
1.
∑
n
=
1
∞
(
−
1
)
n
n
(
9
n
2
−
1
)
=
2
ln
(
2
)
−
3
2
.
∑
n
=
2
∞
1
2
n
[
ζ
(
n
)
−
1
]
=
ln
(
2
)
−
1
2
.
∑
n
=
1
∞
1
2
n
+
1
[
ζ
(
n
)
−
1
]
=
1
−
γ
−
ln
(
2
)
2
.
∑
n
=
1
∞
1
2
2
n
(
2
n
+
1
)
ζ
(
2
n
)
=
1
−
ln
(
2
)
2
.
ln
(
2
)
=
∑
n
=
1
∞
1
2
n
n
.
ln
(
2
)
=
∑
n
=
1
∞
(
1
3
n
+
1
4
n
)
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
.
ln
(
2
)
=
2
3
∑
k
≥
0
1
9
k
(
2
k
+
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
)
)
.
ln
(
2
)
=
∑
n
=
1
∞
1
4
n
2
−
2
n
(γ is the Euler–Mascheroni constant and ζ Riemann's zeta function.)
Some Bailey–Borwein–Plouffe (BBP)-type representations fall also into this category.
∫
0
1
d
x
1
+
x
=
ln
(
2
)
,
or, equivalently,
∫
1
2
d
x
x
=
ln
(
2
)
.
∫
1
∞
d
x
(
1
+
x
2
)
(
1
+
x
)
2
=
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
.
∫
0
∞
1
e
x
−
1
−
2
e
2
x
−
1
d
x
=
ln
(
2
)
.
∫
0
∞
e
−
x
1
−
e
−
x
x
d
x
=
ln
(
2
)
.
∫
0
1
ln
(
x
2
−
1
x
ln
(
x
)
)
d
x
=
−
1
+
ln
(
2
)
+
γ
.
∫
0
π
3
tan
(
x
)
d
x
=
2
∫
0
π
4
tan
(
x
)
d
x
=
ln
(
2
)
.
∫
−
π
4
π
4
ln
(
sin
(
x
)
+
cos
(
x
)
)
d
x
=
−
π
ln
(
2
)
4
.
∫
0
1
x
2
ln
(
1
+
x
)
d
x
=
2
ln
(
2
)
3
−
5
18
.
∫
0
1
x
ln
(
1
+
x
)
ln
(
1
−
x
)
d
x
=
1
4
−
ln
(
2
)
.
∫
0
1
x
3
ln
(
1
+
x
)
ln
(
1
−
x
)
d
x
=
13
96
−
2
ln
(
2
)
3
.
∫
0
1
ln
x
(
1
+
x
)
2
d
x
=
−
ln
(
2
)
.
∫
0
1
ln
(
1
+
x
)
−
x
x
2
d
x
=
1
−
2
ln
(
2
)
.
∫
0
1
d
x
x
(
1
−
ln
(
x
)
)
(
1
−
2
ln
(
x
)
)
=
ln
(
2
)
.
∫
1
∞
ln
(
ln
(
x
)
)
x
3
d
x
=
−
γ
+
ln
(
2
)
2
.
(γ is the Euler–Mascheroni constant.)
The Pierce expansion is A091846
ln
(
2
)
=
1
−
1
1
⋅
3
+
1
1
⋅
3
⋅
12
−
⋯
.
The Engel expansion is A059180
ln
(
2
)
=
1
2
+
1
2
⋅
3
+
1
2
⋅
3
⋅
7
+
1
2
⋅
3
⋅
7
⋅
9
+
⋯
.
The cotangent expansion is A081785
ln
(
2
)
=
cot
(
arccot
(
0
)
−
arccot
(
1
)
+
arccot
(
5
)
−
arccot
(
55
)
+
arccot
(
14187
)
−
⋯
)
.
As an infinite sum of fractions:
ln
(
2
)
=
1
−
1
2
+
1
3
−
1
4
+
1
5
−
⋯
.
It can also be expressed through the Taylor series:
ln
(
2
)
=
1
2
+
1
12
+
1
30
+
1
56
+
1
90
+
⋯
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
,
.
.
.
]
,
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
−
⋱
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
c
=
2
i
3
j
5
k
7
l
⋯
→
ln
(
c
)
=
i
ln
(
2
)
+
j
ln
(
3
)
+
k
ln
(
5
)
+
l
ln
(
7
)
+
⋯
Apart from the logarithms of 2, 3, 5 and 7 shown above, this employs
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.
If ps = qt + d with some small d, then ps/qt = 1 + d/qt and therefore
s
ln
(
p
)
−
t
ln
(
q
)
=
ln
(
1
+
d
q
t
)
=
∑
m
=
1
∞
(
−
1
)
m
+
1
(
d
q
t
)
m
m
.
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
2
ln
(
3
)
=
3
ln
(
2
)
−
∑
k
≥
1
(
−
1
)
k
8
k
k
.
This is actually the third line in the following table of expansions of this type:
Starting from the natural logarithm of q = 10 one might use these parameters:
