NOTES. 99
10 = 1000000000, the number of places, less one, in the result produced by raising 2 to the 1000000000th power will be 301029995. So that reducing these in the ratio of 1000000000, we have log. 10 = 1 and log. 2 =.301029995 &c. The process is explained by Briggs, pages 61-63, and the first steps in the approximation are shown in a tabular form. The table, extended to embrace Napier’s approximation, is given below: in this form it will be found in Hutton’s Introduction to his Mathematical Tables, with further remarks on the subject.
The method, it will be seen, is really one for finding the limits of the logarithm. These limits are carried one place further for each cypher added to the assumed logarithm of 10, but their difference always remains unity in the last place. Bringing together the successive approximations obtained in the table, we find—
When 2 is raised The greater limit of And the less to the power its logarithm is limit is
| When 2 is raised to the power |
| The greater limit of its logarithm is |
| And the less limit is |
| 1 | 1. | 0. | ||
| 10 | .4 | .3 | ||
| 100 | .31 | .30 | ||
| 1000 | .302 | .301 | ||
| 10000 | .3011 | .3010 | ||
| 100000 | .30103 | .30102 | ||
| 1000000 | .301030 | .301029 | ||
| 10000000 | .3010300 | .3010299 | ||
| 100000000 | .30103000 | .30102999 | ||
| 1000000000 | .301029996 | .301029995 |
THE TABLE.
| Powers of a. | Indices of powers of a. |
Number of places in powers of a. | |||
| 24619 | 1000000000 | 1301 | ÷10000 | =log. | 256 |
| 4 | 2 | 1 | ”„ | 4 | |
| 16 | 4 | 2 | ”„ | 16 | |
| 256 | 8 | 3 | ”„ | 256 | |
| 1024 | 10 | 4 | ÷10 | =log. | 2 |
| 10486 etc. | 20 | 7 | ”„ | 4 | |
| 10995 etc.” | 40 | 13 | ”„ | 4 | |
| 12089 etc.” | 80 | 25 | ”„ | 256 | |
