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The decimal value of the natural logarithm of 2 is approximately :\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 In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwe ...
:\log_b 2 = \frac. The
common logarithm In mathematics, the common logarithm is the logarithm with base 10. It is also known as the decadic logarithm and as the decimal logarithm, named after its base, or Briggsian logarithm, after Henry Briggs, an English mathematician who pioneered ...
in particular is () :\log_ 2 \approx 0.301\,029\,995\,663\,981\,195. The inverse of this number is the
binary logarithm In mathematics, the binary logarithm () is the power to which the number must be raised to obtain the value . That is, for any real number , :x=\log_2 n \quad\Longleftrightarrow\quad 2^x=n. For example, the binary logarithm of is , the ...
of 10: : \log_2 10 =\frac \approx 3.321\,928\,095 (). By the Lindemann–Weierstrass theorem, the natural logarithm of any
natural number In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
other than 0 and 1 (more generally, of any positive
algebraic number An algebraic number is a number that is a root of a non-zero polynomial in one variable with integer (or, equivalently, rational) coefficients. For example, the golden ratio, (1 + \sqrt)/2, is an algebraic number, because it is a root of th ...
other than 1) is a
transcendental number In mathematics, a transcendental number is a number that is not algebraic—that is, not the root of a non-zero polynomial of finite degree with rational coefficients. The best known transcendental numbers are and . Though only a few classes ...
.


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.(1+\frac+\frac+\frac+.........) .


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 Zeta (, ; uppercase Ζ, lowercase ζ; grc, ζῆτα, el, ζήτα, label=Demotic Greek, classical or ''zē̂ta''; ''zíta'') is the sixth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 7. It was derived fr ...
= \ln 2. :\sum_^\infty \frac
zeta(n)-1 Zeta (, ; uppercase Ζ, lowercase ζ; grc, ζῆτα, el, ζήτα, label=Demotic Greek, classical or ''zē̂ta''; ''zíta'') is the sixth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 7. It was derived fr ...
= \ln 2 -\frac. :\sum_^\infty \frac
zeta(2n+1)-1 Zeta (, ; uppercase Ζ, lowercase ζ; grc, ζῆτα, el, ζήτα, label=Demotic Greek, classical or ''zē̂ta''; ''zíta'') is the sixth letter of the Greek alphabet. In the system of Greek numerals, it has a value of 7. It was derived fr ...
= 1-\gamma-\frac. :\sum_^\infty \frac\zeta(2n) = 1-\ln 2. ( is the Euler–Mascheroni constant and
Riemann's zeta function The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
.)


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 The Engel expansion of a positive real number ''x'' is the unique non-decreasing sequence of positive integers \ such that :x=\frac+\frac+\frac+\cdots = \frac\left(1+\frac\left(1+\frac\left(1+\cdots\right)\right)\right) For instance, Euler's cons ...
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
integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign ( −1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the languag ...
s is to tabulate the logarithms of the
prime number A prime number (or a prime) is a natural number greater than 1 that is not a Product (mathematics), product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime ...
s and in the next layer the logarithms of the
composite Composite or compositing may refer to: Materials * Composite material, a material that is made from several different substances ** Metal matrix composite, composed of metal and other parts ** Cermet, a composite of ceramic and metallic materials ...
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 (-1)^\frac = \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 In number theory, the Erdős–Moser equation is :1^k+2^k+\cdots+m^k=(m+1)^k, where m and k are positive integers. The only known solution is 11 + 21 = 31, and Paul Erdős conjectured that no further solutions exist. Constraints on solutio ...
: all solutions must come from a convergent of .


References

* * * * * *


External links

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