From the formula for loge (p/q) we may deduce the following very convergent series for loge2, loge3 and loge5, viz.:—
| loge 2 = 2 (7P + 5Q + 3R),
loge 3 = 2 (11P + 8Q + 5R),
loge 5 = 2 (16P + 12Q + 7R),
|
where
| P = | 1 | + 1⁄3 · | 1 | + 1⁄5 · | 1 | + &c. |
| 31 | (31)3 | (31)5 |
| Q = | 1 | + 1⁄3 · | 1 | + 1⁄5 · | 1 | + &c. |
| 49 | (49)3 | (49)5 |
| R = | 1 | + 1⁄3 · | 1 | + 1⁄5 · | 1 | + &c. |
| 161 | (161)3 | (161)5 |
The following still more convenient formulae for the calculation of loge 2, loge 3, &c. were given by J. Couch Adams in the Proc. Roy. Soc., 1878, 27, p. 91. If
| a = log | 10 | = −log ( 1 − | 1 | ), b = log | 25 | = −log ( 1 − | 4 | ), |
| 9 | 10 | 24 | 100 |
| c = log | 81 | = log ( 1 + | 1 | ), d = log | 50 | = −log ( 1 − | 2 | ), |
| 80 | 80 | 49 | 100 |
| e = log | 126 | = log ( 1 + | 8 | ), |
| 125 | 1000 |