log(e)(1 + x) = x - ½x² + (1/3)x³ - ¼x^4 + &c.,
we have, by changing the sign of x,
log(e)(1 - x) = -x - ½x² - (1/3)x³ - ¼x^4 - &c.;
whence
1 + x
log(e) ----- = 2(x + (1/3)x³ + (1/5)x^5 + &c.),
1 - x
p - q
and, therefore, replacing x by -----,
p + q
_ _
p | p - q /p - q\³ /p - q\^5 |
log(e) --- = 2 | ----- + (1/3)( ----- ) + (1/5)( ----- ) + &c. |,
q |_ p + q \p + q/ \p + q/ _|
in which the series is always convergent, so that the formula affords
a method of deducing the logarithm of one number from that of another.
As particular cases we have, by putting q = 1,
_ _
| p - 1 /p - 1\³ /p - 1\^5 |
log(e) p = 2 | ----- + (1/3)( ----- ) + (1/5)( ----- ) + &c. |,
|_ p + 1 \p + 1/ \p + 1/ _|
and by putting q = p + 1,
_ _
| 1 1 1 |
log(e)(p + 1) - log(e)(p) = 2 | ------ + (1/3)--------- + (1/5)----------- + &c. |;
|_ 2p + 1 (2p + 1)³ (2p + 1)^5 _|
the former of these equations gives a convergent series for log(e)p,
and the latter a very convergent series by means of which the
logarithm of any number may be deduced from the logarithm of the
preceding number.
From the formula for log(e)(p/q) we may deduce the following very
convergent series for log(e)2, log(e)3 and log(e)5, viz.:--
log(e)2 = 2( 7P + 5Q + 3R),
log(e)3 = 2(11P + 8Q + 5R),
log(e)5 = 2(16P + 12Q + 7R),
where
1 1 1
P = -- + (1/3) · ------ + (1/5) · ------ + &c.
31 (31)^3 (31)^5
1 1 1
Q = -- + (1/3) · ------ + (1/5) · ------ + &c.
49 (49)^3 (49)^5
1 1 1
R = --- + (1/3) · ------- + (1/5) · ------- + &c.
161 (161)^3 (161)^5
The following still more convenient formulae for the calculation of
log(e)2, log(e)3, &c. were given by J. Couch Adams in the _Proc. Roy.
Soc._, 1878, 27, p. 91. If
10 / 1 \ 25 / 4 \
a = log -- = -log ( 1 - -- ), b = log -- = -log ( 1 - --- ),
9 \ 10 / 24 \ 100 /
81 / 1 \ 50 / 2 \
c = log -- = log ( 1 + -- ), d = log -- = -log ( 1 - --- ),
80 \ 80 / 49 \ 100 /
126 / 8 \
e = log --- = log ( 1 + ---- ),
125 \ 1000 /
then
log 2 = 7a - 2b + 3c, log 3 = 11a - 3b + 5c, log 5 = 16a - 4b + 7c,
and
log 7 = ½(39a - 10b + 17c - d) or = 19a - 4b + 8c + e,
and we have the equation of condition,
a - 2b + c = d + 2e.
By means of these formulae Adams calculated the values of log(e)2,
log(e)3, log(e)5, and log(e)7 to 276 places of decimals, and deduced
the value of log(e)10 and its reciprocal M, the modulus of the
Briggian system of logarithms. The value of the modulus found by Adams
is
Mo = 0.43429 44819 03251 82765 11289
18916 60508 22943 97005 80366
65661 14453 78316 58646 49208
87077 47292 24949 33843 17483
18706 10674 47663 03733 64167
92871 58963 90656 92210 64662
81226 58521 27086 56867 03295
93370 86965 88266 88331 16360
77384 90514 28443 48666 76864
65860 85135 56148 21234 87653
43543 43573 17253 83562 21868
25
which is true certainly to 272, and probably to 273, places (_Proc.
Roy. Soc._, 1886, 42, p. 22, where also the values of the other
logarithms are given).
If the logarithms are to be Briggian all the series in the preceding
formulae must be multiplied by M, the modulus; thus,
log(10) (1 + x) = M (x - ½x² + (1/3)x³ - ¼x^4 + &c.),
and so on.
As has been stated, Abraham Sharp's table contains 61-decimal
Briggian logarithms of primes up to 1100, so that the logarithms of
all composite numbers whose greatest prime factor does not exceed this
number may be found by simple addition; and Wolfram's table gives
48-decimal hyperbolic logarithms of primes up to 10,009. By means of
these tables and of a factor table we may very readily obtain the
Briggian logarithm of a number to 61 or a less number of places or of
its hyperbolic logarithm to 48 or a less number of places in the
following manner. Suppose the hyperbolic logarithm of the prime number
43,867 required. Multiplying by 50, we have 50 × 43,867 = 2,193,350,
and on looking in Burckhardt's _Table des diviseurs_ for a number near
to this which shall have no prime factor greater than 10,009, it
appears that
2,193,349 = 23 × 47 × 2029;
thus
43,867 = (1/50)(23 × 47 × 2029 + 1),
and therefore
log(e) 43,867 = log(e) 23 + log(e) 47 + log(e) 2029 - log(e) 50
1 1 1
+ --------- - ½ ------------ + (1/3) ---------- - &c.
2,193,349 (2,193,349)² (193,349)³