The natural logarithm of 2
The natural logarithm of 2 is the time constant of doubling, and the sum 1 − 1/2 + 1/3 − 1/4 + …. Its value is 0.69314718055994530941723212145817656807550013436025…, irrational and transcendental.
How to work it out
- 1
What it is: the time constant of doubling, and the sum 1 − 1/2 + 1/3 − 1/4 + ….
- 2
The first 100 digits, computed exactly with big-integer arithmetic:
- 3
0.69314718 0559945309 4172321214 5817656807 5500134360 2552541206 8000949339 3621969694 7156058633 2699641868 75
- 4
Its continued fraction is [6; 1; 13; 1; 1; 2; 4; 1; 22; 1, …]. Cutting it off gives the best fractions for a given size of denominator, such as 7/1, 97/14, 104/15.
LaTeX source of the formulas on this page
\[ \ln 2 = 0.693147180559\ldots \] \[ \ln 2 \]
How often each digit appears in the first 100 digits (computed)
| Digit | Count |
|---|---|
| 0 | 11 |
| 1 | 11 |
| 2 | 8 |
| 3 | 10 |
| 4 | 9 |
| 5 | 13 |
| 6 | 13 |
| 7 | 6 |
| 8 | 7 |
| 9 | 12 |
Best fractions from the continued fraction (computed)
| Fraction | Decimal | Error |
|---|---|---|
| 6/1 | 6.00000000000 | 5.31e+0 |
| 7/1 | 7.00000000000 | 6.31e+0 |
| 97/14 | 6.92857142857 | 6.24e+0 |
| 104/15 | 6.93333333333 | 6.24e+0 |
| 201/29 | 6.93103448276 | 6.24e+0 |
| 506/73 | 6.93150684932 | 6.24e+0 |
| 2225/321 | 6.93146417445 | 6.24e+0 |
| 2731/394 | 6.93147208122 | 6.24e+0 |
Questions
What is the value of The natural logarithm of 2?
0.69314718055994530941723212145817656807550013436025…
Is The natural logarithm of 2 irrational?
It is irrational and transcendental.
What are the first 10 digits of The natural logarithm of 2?
0.6931471805.
What is a good fraction for The natural logarithm of 2?
7/1 is simple, and 104/15 is accurate to about 6.2e+0.