A016628 Decimal expansion of log(5).

1, 6, 0, 9, 4, 3, 7, 9, 1, 2, 4, 3, 4, 1, 0, 0, 3, 7, 4, 6, 0, 0, 7, 5, 9, 3, 3, 3, 2, 2, 6, 1, 8, 7, 6, 3, 9, 5, 2, 5, 6, 0, 1, 3, 5, 4, 2, 6, 8, 5, 1, 7, 7, 2, 1, 9, 1, 2, 6, 4, 7, 8, 9, 1, 4, 7, 4, 1, 7, 8, 9, 8, 7, 7, 0, 7, 6, 5, 7, 7, 6, 4, 6, 3, 0, 1, 3, 3, 8, 7, 8, 0, 9, 3, 1, 7, 9, 6, 1

Offset

1,2

References

Milton Abramowitz and Irene A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 2.

Horace S. Uhler, 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, (1940). 205-212.

Links

Harry J. Smith, Table of n, a(n) for n = 1..20000

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

Simon Plouffe, The natural logarithm of 5 to 10000 digits

Index entries for transcendental numbers

Formula

From Peter Bala, Nov 11 2019: (Start)

log(5) = 2*sqrt(2)*Integral_{t = 0..sqrt(2)/2} (1 - t^2)/(1 + t^4) dt.

log(5) = Sum_{n >= 0} (4*n+5)/((4*n+1)*(4*n+3))*(-1/4)^n.

log(5) = (1/4)*Sum_{n >= 0} ( 8/(8*n+1) - 4/(8*n+3) - 2/(8*n+5) + 1/(8*n+7) )*(1/16)^n, a BBP-type formula. (End)

log(5) = 2*Sum_{n >= 0} (-1)^(n*(n+1)/2)*1/((2*n+1)*2^n). - Peter Bala, Oct 29 2020

log(5) = Integral_{x = 0..1} (x^4 - 1)/log(x) dx. - Peter Bala, Nov 14 2020

log(5) = 2*Sum_{n >= 1} 1/(n*P(n, 3/2)*P(n-1, 3/2)), where P(n, x) denotes the n-th Legendre polynomial. The first 20 terms of the series gives the approximation log(5) = 1.6094379124341003(29...), correct to 16 decimal places. - Peter Bala, Mar 18 2024

Example

1.60943791243410037460075933322618763952560135426851772191264789... - Harry J. Smith, May 16 2009

Mathematica

RealDigits[Log[5], 10, 125][[1]] (* Alonso del Arte, Oct 04 2014 *)

Prog

(PARI) default(realprecision, 20080); x=log(5); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b016628.txt", n, " ", d)); \\ Harry J. Smith, May 16 2009

Crossrefs

Cf. A016733 (continued fraction). - Harry J. Smith, May 16 2009

Sequence in context: A323606 A133077 A200124 * A112246 A189037 A153201

Adjacent sequences: A016625 A016626 A016627 * A016629 A016630 A016631

Keyword

nonn,cons,easy

Author

N. J. A. Sloane

Status

approved