The decimal value of the
natural logarithm of
2
is approximately
:
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 ...
:
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 ()
:
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:
:
().
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
:
This is the well-known "
alternating harmonic series".
:
:
:
:
:
:
Binary rising constant factorial
:
:
:
:
:
:
Other series representations
:
:
:
:
:
:
:
:
:
:
using
:
(sums of the reciprocals of
decagonal numbers)
Involving the Riemann Zeta function
:
:
:
:
( 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
:
(See more about
Bailey–Borwein–Plouffe (BBP)-type representations.)
Applying the three general series for natural logarithm to 2 directly gives:
:
:
:
Applying them to
gives:
:
:
:
Applying them to
gives:
:
:
:
Applying them to
gives:
:
:
:
Representation as integrals
The natural logarithm of 2 occurs frequently as the result of integration. Some explicit formulas for it include:
:
:
:
:
:
Other representations
The Pierce expansion is
:
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
:
The cotangent expansion is
:
The simple
continued fraction expansion is
: