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Natural Logarithm of 2

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The natural logarithm of 2 is a transcendental quantity that arises often in decay problems, especially when half-lives are being converted to decay constants. ln2 has numerical value

ln2=0.69314718055994530941...
(1)

(OEIS A002162).

The irrationality measure of ln2 is known to be less than 3.8913998 (Rukhadze 1987, Hata 1990).

It is not known if ln2 is normal (Bailey and Crandall 2002).

The alternating series and BBP-type formula

eta(1)=sum_(k=1)^infty((-1)^(k-1))/k=ln2
(2)

converges to the natural logarithm of 2, where eta(x) is the Dirichlet eta function. This identity follows immediately from setting x=1 in the Mercator series, yielding

ln2=sum_(k=1)^infty((-1)^(k+1))/k.
(3)

It is also a special case of the identity

1/nsum_(k=1)^n(-1)^(k-1)n/k=ln2-(-1)^nPhi(-1,1,n+1),
(4)

where Phi(z,s,a) is the Lerch transcendent.

This is the simplest in an infinite class of such identities, the first few of which are

ln2=sum_(k=0)^(infty)(-1)^k(1/(3k+1)-1/(3k+2)+1/(3k+3))
(5)
=sum_(k=0)^(infty)(-1)^k(1/(5k+1)-1/(5k+2)+1/(5k+3)-1/(5k+4)+1/(5k+5))
(6)

(E. W. Weisstein, Oct. 7, 2007).

There are many other classes of BBP-type formulas for ln2, including

ln2=1/3sum_(k=0)^(infty)1/((-27)^k)(3/(6k+1)-2/(6k+3)-1/(6k+4))
(7)
=1/6sum_(k=0)^(infty)1/((-27)^k)(-3/(6k+1)+9/(6k+2)+8/(6k+3)+1/(6k+4)-1/(6k+5))
(8)
=sum_(k=0)^(infty)1/((-19683)^k)((2187)/(18k+1)-(1458)/(18k+3)-(729)/(18k+4)-(81)/(18k+7)+(54)/(18k+9)+(27)/(18k+10)+3/(18k+13)-2/(18k+15)-1/(18k+16))
(9)
=1/2sum_(k=0)^(infty)1/((-4)^k)(2/(4k+1)-1/(4k+3)-1/(4k+4))
(10)
=1/8sum_(k=0)^(infty)1/((-8)^k)(8/(3k+1)-4/(3k+2)-1/(3k+3)).
(11)

Plouffe (2006) found the beautiful sum

ln2=10sum_(n=1)^infty1/(n(e^(pin)+1))+6sum_(n=1)^infty1/(n(e^(pin)-1)) -4sum_(n=1)^infty1/(n(e^(2pin)+1)).
(12)

A rapidly converging Zeilberger-type sum due to A. Lupas is given by

ln2=3/4-1/8sum_(n=1)^infty(2n; n)((-1)^(n-1)(5n+1))/(16^nn(n+1/2))
(13)

(Lupas 2000; typos corrected).

The following integral is given in terms of ln2,

int_2^infty(dx)/(xln^2x)=1/(ln2).
(14)
NaturalLogOf2

The plot above shows the result of truncating the series for ln2 after n terms.

Taking the partial series gives the analytic result

sum_(k=1)^(N)((-1)^(k+1))/k=ln2+1/2(-1)^N[psi_0(1/2(N+1))-psi_0(1+1/2N)]
(15)
=ln2+1/2(-1)^N[H_((N-1)/2)-H_(N/2)],
(16)

where psi_0(z) is the digamma function and H_n is a harmonic number. Rather amazingly, expanding about infinity gives the series

sum_(k=1)^N((-1)^(k+1))/k=ln2+(-1)^N[1/(2N)+sum_(k=0)^infty((-1)^kT_k)/(4^kN^(2k))]
(17)

(Borwein and Bailey 2002, p. 50), where T_n is a tangent number. This means that truncating the series for ln2 at half a large power of 10 can give a decimal expansion for ln2 whose decimal digits are largely correct, but where wrong digits occur with precise regularity.

Ln2TangentNumbers

For example, taking N=5×10^6 gives a decimal value equal to the second row of digits above, where the sequence of differences from the decimal digits of ln2 in the top row is precisely the tangent numbers with alternating signs (Borwein and Bailey 2002, p. 49).

Beautiful BBP-type formulas for ln2 are given by

ln2=1/2sum_(k=0)^(infty)1/(2^k)1/(k+1)
(18)
=sum_(k=1)^(infty)1/(k·2^k)
(19)

(Bailey et al. 2007, p. 31) and

ln2=2/3sum_(k=0)^infty1/(9^k(2k+1))
(20)

(Borwein and Bailey 2002, p. 129).

A BBP-type formula for (ln2)^2 discovered using the PSLQ algorithm is

(ln2)^2=1/(32)sum_(k=0)^infty1/(64^k)[(64)/((6k+1)^2)-(160)/((6k+2)^2)-(56)/((6k+3)^2)-(40)/((6k+4)^2)+4/((6k+5)^2)-1/((6k+6)^2)]
(21)

(Bailey and Plouffe 1997; Borwein and Bailey 2002, p. 128).

The sum

sum_(k=2^n)^(2^(n+1)-1)1/k=psi_0(2^(n+1))-psi_0(2^n)
(22)

has the limit

lim_(n->infty)sum_(k=2^n)^(2^(n+1)-1)1/k=ln2
(23)

(Borwein et al. 2004, p. 10).

See also

Alternating Harmonic Series, Dirichlet Eta Function, Mercator Series, Natural Logarithm, Natural Logarithm of 2 Continued Fraction, Natural Logarithm of 2 Digits, q-Harmonic Series

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References

Bailey, D. H.; Borwein, J. M.; Calkin, N. J.; Girgensohn, R.; Luke, D. R.; and Moll, V. H. "Integer Relation Detection." §2.2 in Experimental Mathematics in Action. Wellesley, MA: A K Peters, pp. 29-31, 2007.Bailey, D. H.; Borwein, P. B.; and Plouffe, S. "On the Rapid Computation of Various Polylogarithmic Constants." Math. Comput. 66, 903-913, 1997.Bailey, D. H. and Crandall, R. E. "Random Generators and Normal Numbers." Exper. Math. 11, 527-546, 2002.Bailey, D. and Plouffe, S. "Recognizing Numerical Constants." Organic Mathematics. Proceedings of the Workshop Held in Burnaby, BC, December 12-14, 1995 (Ed. J. Borwein, P. Borwein, L. Jörgenson, and R. Corless). Providence, RI: Amer. Math. Soc., pp. 73-88, 1997. http://www.cecm.sfu.ca/organics/papers/bailey/.Borwein, J. and Bailey, D. Mathematics by Experiment: Plausible Reasoning in the 21st Century. Wellesley, MA: A K Peters, 2003.Borwein, J.; Bailey, D.; and Girgensohn, R. Experimentation in Mathematics: Computational Paths to Discovery. Wellesley, MA: A K Peters, 2004.Gourdon, X. and Sebah, P. "The Constant ln2." http://numbers.computation.free.fr/Constants/Log2/log2.html.Hata, M. "Legendre Type Polynomials and Irrationality Measures." J. reine angew. Math. 407, 99-125, 1990.Huylebrouck, D. "Similarities in Irrationality Proofs for pi, ln2, zeta(2), and zeta(3)." Amer. Math. Monthly 108, 222-231, 2001.Lupas, A. "Formulae for Some Classical Constants." In Proceedings of ROGER-2000. 2000. http://www.lacim.uqam.ca/~plouffe/articles/alupas1.pdf.Plouffe, S. "Identities Inspired from Ramanujan Notebooks (Part 2)." Apr. 2006. http://www.lacim.uqam.ca/~plouffe/inspired2.pdf.Rukhadze, E. A. "A Lower Bound for the Rational Approximation of ln2 by Rational Numbers." Vestnik Moskov Univ. Ser. I Math. Mekh., No. 6, 25-29 and 97, 1987. [Russian].Sloane, N. J. A. Sequences A002162/M4074, A016730, and A059180 in "The On-Line Encyclopedia of Integer Sequences."Sweeney, D. W. "On the Computation of Euler's Constant." Math. Comput. 17, 170-178, 1963.Uhler, H. S. "Recalculation and Extension of the Modulus and of the Logarithms of 2, 3, 5, 7 and 17." Proc. Nat. Acad. Sci. U.S.A. 26, 205-212, 1940.

Cite this as:

Weisstein, Eric W. "Natural Logarithm of 2." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/NaturalLogarithmof2.html

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