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

Author

N. J. A. Sloane

Keywords

• nonn
• cons
• easy

Examples

			1.60943791243410037460075933322618763952560135426851772191264789... - _Harry J. Smith_, May 16 2009
		

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

Crossrefs

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

Programs

Mathematica

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

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

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