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Calculating Logarithms with a Series

​
argument
1
number of terms
7
decimal places
8
a
a
1
100000000
b
a
9
1
9
11111111
c
b
9
1
81
1234567
d
c
9
1
729
137174
e
d
9
1
6561
15241
f
e
9
1
59049
1693
g
f
9
1
531441
188
h
g
9
1
4782969
20
A
a
1
100000000
B
b
3
1
27
3703703
C
c
5
1
405
246913
D
d
7
1
5103
19596
E
e
9
1
59049
1693
F
f
11
1
649539
153
G
g
13
1
6908733
14
H
h
15
1
71744535
1
S
103972073
2
3
S
69314715
exact
69314718
This series was often used to calculate logarithmic tables:
log(n+1)-log(n)=
2
2n+1
1+
1
3
·
1
2
(2n+1)
+
1
5
·
1
4
(2n+1)
+…
.
For
n=1
, this gives
log(2)
. To calculate
log(3)
, add
log(2)
to the series with
n=2
, and so on.
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