Natural logarithm of 2

Natural logarithm of 2

The natural logarithm of 2 as an area under the curve 1/x.
Rationality	Irrational
Representations
Decimal	0.6931471805599453094...

In mathematics, the natural logarithm of 2 is the unique real number argument such that the exponential function equals two. It appears regularly in various formulas and is also given by the alternating harmonic series. The decimal value of the natural logarithm of 2 (sequence A002162 in the OEIS) truncated at 30 decimal places is given by:

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

The logarithm of 2 in other bases is obtained with the formula

${\displaystyle \log _{b}2={\frac {\ln 2}{\ln b}}.}$

The common logarithm in particular is (OEIS: A007524)

${\displaystyle \log _{10}2\approx 0.301\,029\,995\,663\,981\,195.}$

The inverse of this number is the binary logarithm of 10:

${\displaystyle \log _{2}10={\frac {1}{\log _{10}2}}\approx 3.321\,928\,095}$ (OEIS: A020862).

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.