natural logarithm of 2

In mathematics, the natural logarithm of 2 is the unique real number argument such that the exponential function equals two. It appears frequently in various formulas and is also given by the alternating harmonic series. The decimal value of the natural logarithm of 2 truncated at 30 decimal places is given by:

:$\ln 2 \approx 0.693\,147\,180\,559\,945\,309\,417\,232\,121\,458.$

The logarithm of 2 in other bases is obtained with the formula
:$\log_b 2 = \frac.$
The common logarithm in particular is ()
:$\log_ 2 \approx 0.301\,029\,995\,663\,981\,195.$
The inverse of this number is the binary logarithm of 10:
:$\log_2 10 =\frac \approx 3.321\,928\,095$ ().

By the Lindemann–Weierstrass theorem, the natural logarithm of any natural number other than 0 and 1 (more generally, of any positive algebraic number other than 1) is a transcendental number. It is also contained in the ring of algebraic periods.

Series representations

Rising alternate factorial

:$\ln 2 = \sum_^\infty \frac=1-\frac12+\frac13-\frac14+\frac15-\frac16+\cdots.$ This is the well-known "alternating harmonic series".
:$\ln 2 = \frac +\frac\sum_^\infty \frac.$
:$\ln 2 = \frac +\frac\sum_^\infty \frac.$
:$\ln 2 = \frac +\frac\sum_^\infty \frac.$
:$\ln 2 = \frac +\frac\sum_^\infty \frac.$
:$\ln 2 = \frac +\frac\sum_^\infty \frac.$
:$\ln 2 = \frac\left(1+\frac+\frac+\frac+\dots\right) .$

Binary rising constant factorial

:$\ln 2 = \sum_^\infty \frac.$
:$\ln 2 = 1 -\sum_^\infty \frac.$
:$\ln 2 = \frac + 2 \sum_^\infty \frac .$
:$\ln 2 = \frac - 6 \sum_^\infty \frac .$
:$\ln 2 = \frac + 24 \sum_^\infty \frac .$
:$\ln 2 = \frac - 120 \sum_^\infty \frac .$

Other series representations

:$\sum_^\infty \frac = \ln 2.$
:$\sum_^\infty \frac = 2\ln 2 -1.$
:$\sum_^\infty \frac = \ln 2 -1.$
:$\sum_^\infty \frac = 2\ln 2 -\frac.$
:$\sum_^\infty \frac = \ln 2.$
:$\sum_^\infty \frac = \ln 2.$
:$\sum_^\infty \frac = \frac+\frac.$
:$\sum_^\infty \frac = -\frac+\frac.$
:$\sum_^\infty \frac = \frac.$
:$\sum_^\infty \frac = 18 - 24 \ln 2$ using $\lim_ \sum_^ \frac = \ln 2$
:$\sum_^\infty \frac = \ln 2 + \frac$ (sums of the reciprocals of decagonal numbers)

Involving the Riemann Zeta function

:\sum_^\infty \frac zeta(2n)-1= \ln 2.
:\sum_^\infty \frac zeta(n)-1= \ln 2 -\frac.
:\sum_^\infty \frac zeta(2n+1)-1= 1-\gamma-\frac.
:$\sum_^\infty \frac\zeta(2n) = 1-\ln 2.$

( is the Euler–Mascheroni constant and Riemann's zeta function.)

BBP-type representations

:$\ln 2 = \frac + \frac \sum_^\infty \left(\frac+\frac+\frac+\frac\right) \frac .$
(See more about Bailey–Borwein–Plouffe (BBP)-type representations.)

Applying the three general series for natural logarithm to 2 directly gives:
:$\ln 2 = \sum_^\infty \frac.$
:$\ln 2 = \sum_^\infty \frac.$
:$\ln 2 = \frac \sum_^\infty \frac.$

Applying them to $\textstyle 2 = \frac \cdot \frac$ gives:
:$\ln 2 = \sum_^\infty \frac + \sum_^\infty \frac .$
:$\ln 2 = \sum_^\infty \frac + \sum_^\infty \frac .$
:$\ln 2 = \frac \sum_^\infty \frac + \frac \sum_^\infty \frac .$

Applying them to $\textstyle 2 = (\sqrt)^2$ gives:
:$\ln 2 = 2 \sum_^\infty \frac .$
:$\ln 2 = 2 \sum_^\infty \frac .$
:$\ln 2 = \frac \sum_^\infty \frac .$

Applying them to $\textstyle 2 = ^ \cdot ^ \cdot ^$ gives:
:$\ln 2 = 7 \sum_^\infty \frac + 3 \sum_^\infty \frac + 5 \sum_^\infty \frac .$
:$\ln 2 = 7 \sum_^\infty \frac + 3 \sum_^\infty \frac + 5 \sum_^\infty \frac .$
:$\ln 2 = \frac \sum_^\infty \frac + \frac \sum_^\infty \frac + \frac \sum_^\infty \frac .$

Representation as integrals

The natural logarithm of 2 occurs frequently as the result of integration. Some explicit formulas for it include:

:$\int_0^1 \frac = \int_1^2 \frac = \ln 2$

:$\int_0^\infty e^\frac \, dx= \ln 2$

:$\int_0^\infty 2^ dx= \frac$

:$\int_0^\frac \tan x \, dx=2\int_0^\frac \tan x \, dx = \ln 2$

:$-\frac\int_^ \frac \, dx= \ln 2$

Other representations

The Pierce expansion is
:$\ln 2 = 1 -\frac+\frac -\cdots.$
The Engel expansion is
:$\ln 2 = \frac + \frac + \frac + \frac+\cdots.$
The cotangent expansion is
:$\ln 2 = \cot().$
The simple continued fraction expansion is
: \ln 2 = \left 0; 1, 2, 3, 1, 6, 3, 1, 1, 2, 1, 1, 1, 1, 3, 10, 1, 1, 1, 2, 1, 1, 1, 1, 3, 2, 3, 1,...\right/math>,
which yields rational approximations, the first few of which are 0, 1, 2/3, 7/10, 9/13 and 61/88.

This generalized continued fraction:
: \ln 2 = \left 0;1,2,3,1,5,\tfrac,7,\tfrac,9,\tfrac,...,2k-1,\frac,...\right,
:also expressible as
:$\ln 2 = \cfrac = \cfrac$

Bootstrapping other logarithms

Given a value of , a scheme of computing the logarithms of other integers is to tabulate the logarithms of the prime numbers and in the next layer the logarithms of the composite numbers based on their factorizations
:$c=2^i3^j5^k7^l\cdots\rightarrow \ln(c)=i\ln(2)+j\ln(3)+k\ln(5)+l\ln(7)+\cdots$
This employs

In a third layer, the logarithms of rational numbers are computed with , and logarithms of roots via .

The logarithm of 2 is useful in the sense that the powers of 2 are rather densely distributed; finding powers close to powers of other numbers is comparatively easy, and series representations of are found by coupling 2 to with logarithmic conversions.

Example

If with some small , then and therefore
:$s\ln p -t\ln q = \ln\left(1+\frac\right) = \sum_^\infty \frac\left(\frac\right)^m = \sum_^\infty \frac ^ .$
Selecting represents by and a series of a parameter that one wishes to keep small for quick convergence. Taking , for example, generates
:$2\ln 3 = 3\ln 2 -\sum_\frac = 3\ln 2 + \sum_^\infty \frac ^ .$
This is actually the third line in the following table of expansions of this type:

Starting from the natural logarithm of one might use these parameters:

Known digits

This is a table of recent records in calculating digits of . As of December 2018, it has been calculated to more digits than any other natural logarithm of a natural number, except that of 1.

See also

* Rule of 72#Continuous compounding, in which figures prominently
* Half-life#Formulas for half-life in exponential decay, in which figures prominently
* Erdős–Moser equation: all solutions must come from a convergent of .

References

*
*
*
*
*
*

External links

*
*

{{DEFAULTSORT:Natural Logarithm Of 2
Logarithms
Mathematical constants
Real transcendental numbers