Logaritma alami dari 2
Bootstrap logaritma lainnya > Contoh

Jika  p^{s}=q^{t}+d dengan beberapa  d, maka  {\frac {p^{s}}{q^{t}}}=1+{\frac {d}{q^{t}}} dan karena itu
Memilih  q=2 mewakili  \ln(p) oleh  \ln 2 dan sebuah deret dari sebuah parameter  {\frac {d}{q^{t}}} yang ingin tetap kecil untuk konvergen cepat. Mengambil  3^{2}=2^{3}+1, sebagai contoh, menghasilkan
Ini sebenarnya baris ketiga dalam tabel ekspansi tipe ini:
|  s |  p |  t |  q |  {\frac {d}{q^{t}}} |
| --- | --- | --- | --- | --- |
| 1 | 3 | 1 | 2 |  {\frac {1}{2}}=0.500\,000\,00\dots  |
| 1 | 3 | 2 | 2 |  -{\frac {1}{4}}=-0.250\,000\,00\dots  |
| 2 | 3 | 3 | 2 |  {\frac {1}{8}}=0.125\,000\,00\dots  |
| 5 | 3 | 8 | 2 |  -{\frac {13}{256}}=-0.050\,781\,25\dots  |
| 12 | 3 | 19 | 2 |  {\frac {7153}{524288}}=0.013\,643\,26\dots  |
| 1 | 5 | 2 | 2 |  {\frac {1}{4}}=0.250\,000\,00\dots  |
| 3 | 5 | 7 | 2 |  -{\frac {3}{128}}=-0.023\,437\,50\dots  |
| 1 | 7 | 2 | 2 |  {\frac {3}{4}}=0.750\,000\,00\dots  |
| 1 | 7 | 3 | 2 |  -{\frac {1}{8}}=-0.125\,000\,00\dots  |
| 5 | 7 | 14 | 2 |  {\frac {423}{16384}}=0.025\,817\,87\dots  |
| 1 | 1 | 3 | 2 |  {\frac {3}{8}}=0.375\,000\,00\dots  |
| 2 | 11 | 7 | 2 |  -{\frac {7}{128}}=-0.054\,687\,50\dots  |
| 11 | 11 | 38 | 2 |  {\frac {10433763667}{274877906944}}=0.037\,957\,81\dots  |
| 1 | 13 | 3 | 2 |  {\frac {5}{8}}=0.625\,000\,00\dots  |
| 1 | 13 | 4 | 2 |  -{\frac {3}{16}}=-0.187\,500\,00\dots  |
| 3 | 13 | 11 | 2 |  {\frac {149}{2048}}=0.072\,753\,91\dots  |
| 7 | 13 | 26 | 2 |  -{\frac {4360347}{67108864}}=-0.064\,974\,23\dots  |
| 10 | 13 | 37 | 2 |  {\frac {419538377}{137438953472}}=0.003\,052\,54\dots  |
| 1 | 17 | 4 | 2 |  {\frac {1}{16}}=0.062\,500\,00\dots  |
| 1 | 19 | 4 | 2 |  {\frac {3}{16}}=0.187\,500\,00\dots  |
| 4 | 19 | 17 | 2 |  -{\frac {751}{131072}}=-0.005\,729\,68\dots  |
| 1 | 23 | 4 | 2 |  {\frac {7}{16}}=0.437\,500\,00\dots  |
| 1 | 23 | 5 | 2 |  -{\frac {9}{32}}=-0.281\,250\,00\dots  |
| 2 | 23 | 9 | 2 |  {\frac {17}{512}}=0.033\,203\,12\dots  |
| 1 | 29 | 4 | 2 |  {\frac {13}{16}}=0.812\,500\,00\dots  |
| 1 | 29 | 5 | 2 |  -{\frac {3}{32}}=-0.093\,750\,00\dots  |
| 7 | 29 | 34 | 2 |  {\frac {70007125}{17179869184}}=0.004\,074\,95\dots  |
| 1 | 31 | 5 | 2 |  -{\frac {1}{32}}=-0.031\,250\,00\dots  |
| 1 | 37 | 5 | 2 |  {\frac {5}{32}}=0.156\,250\,00\dots  |
| 4 | 37 | 21 | 2 |  -{\frac {222\,991}{2\,097\,152}}=-0.106\,330\,39\dots  |
| 5 | 37 | 26 | 2 |  {\frac {2\,235\,093}{67\,108\,864}}=0.033\,305\,48\dots  |
| 1 | 41 | 5 | 2 |  {\frac {9}{32}}=0.281\,250\,00\dots  |
| 2 | 41 | 11 | 2 |  -{\frac {367}{2048}}=-0.179\,199\,22\dots  |
| 3 | 41 | 16 | 2 |  {\frac {3385}{65\,536}}=0.051\,651\,00\dots  |
| 1 | 43 | 5 | 2 |  {\frac {11}{32}}=0.343\,750\,00\dots  |
| 2 | 43 | 11 | 2 |  -{\frac {199}{2048}}=-0.097\,167\,97\dots  |
| 5 | 43 | 27 | 2 |  {\frac {12\,790\,715}{134\,217\,728}}=0.095\,298\,25\dots  |
| 7 | 43 | 38 | 2 |  {\frac {3\,059\,295\,837}{274\,877\,906\,944}}=0.011\,129\,65\dots  |
Dimulai dari logaritma natural dari  q=10, salah satunya dapat menggunakan parameter-parameter ini:
|  s |  p |  t |  q |  {\frac {d}{q^{t}}} |
| --- | --- | --- | --- | --- |
| 10 | 2 | 3 | 10 |  {\frac {3}{125}}=0.024\,000\,00\dots  |
| 21 | 3 | 10 | 10 |  {\frac {460\,353\,203}{10\,000\,000\,000}}=0.046\,035\,32\dots  |
| 3 | 5 | 2 | 10 |  {\frac {1}{4}}=0.250\,000\,\dots  |
| 10 | 5 | 7 | 10 |  -{\frac {3}{128}}=-0.023\,437\,50\dots  |
| 6 | 7 | 5 | 10 |  {\frac {17\,649}{100\,000}}=0.176\,490\,00\dots  |
| 13 | 7 | 11 | 10 |  -{\frac {3\,110\,989\,593}{100\,000\,000\,000}}=-0.031\,109\,90\dots  |
| 1 | 11 | 1 | 10 |  {\frac {1}{10}}=0.100\,000\,00\dots  |
| 1 | 13 | 1 | 10 |  {\frac {3}{10}}=0.300\,000\,00\dots  |
| 8 | 13 | 9 | 10 |  -{\frac {184\,269\,279}{1\,000\,000\,000}}=-0.184\,269\,28\dots  |
| 9 | 13 | 10 | 10 |  {\frac {604\,499\,373}{10\,000\,000\,000}}=0.060\,449\,94\dots  |
| 1 | 17 | 1 | 10 |  {\frac {7}{10}}=0.700\,000\,00\dots  |
| 4 | 17 | 5 | 10 |  -{\frac {16\,479}{100\,000}}=-0.164\,790\,00\dots  |
| 9 | 17 | 11 | 10 |  {\frac {18\,587\,876\,497}{100\,000\,000\,000}}=0.185\,878\,76\dots  |
| 3 | 19 | 4 | 10 |  -{\frac {3141}{10\,000}}=-0.314\,100\,00\dots  |
| 4 | 19 | 5 | 10 |  {\frac {30\,321}{100\,000}}=0.303\,210\,00\dots  |
| 7 | 19 | 9 | 10 |  -{\frac {106\,128\,261}{1\,000\,000\,000}}=-0.106\,128\,26\dots  |
| 2 | 23 | 3 | 10 |  -{\frac {471}{1000}}=-0.471\,000\,00\dots  |
| 3 | 23 | 4 | 10 |  {\frac {2167}{10\,000}}=0.216\,700\,00\dots  |
| 2 | 29 | 3 | 10 |  -{\frac {159}{1000}}=-0.159\,000\,00\dots  |
| 2 | 31 | 3 | 10 |  {\frac {39}{1000}}=-0.039\,000\,00\dots  |