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
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
KEYWORD
AUTHOR
STATUS
approved