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%I #48 Mar 21 2024 06:23:45
%S 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,
%T 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,
%U 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
%N Decimal expansion of log(5).
%D 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.
%D 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.
%H Harry J. Smith, <a href="/A016628/b016628.txt">Table of n, a(n) for n = 1..20000</a>
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H Simon Plouffe, <a href="http://www.plouffe.fr/simon/constants/log5.txt">The natural logarithm of 5 to 10000 digits</a>
%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>
%F From _Peter Bala_, Nov 11 2019: (Start)
%F log(5) = 2*sqrt(2)*Integral_{t = 0..sqrt(2)/2} (1 - t^2)/(1 + t^4) dt.
%F log(5) = Sum_{n >= 0} (4*n+5)/((4*n+1)*(4*n+3))*(-1/4)^n.
%F 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)
%F log(5) = 2*Sum_{n >= 0} (-1)^(n*(n+1)/2)*1/((2*n+1)*2^n). - _Peter Bala_, Oct 29 2020
%F log(5) = Integral_{x = 0..1} (x^4 - 1)/log(x) dx. - _Peter Bala_, Nov 14 2020
%F 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
%e 1.60943791243410037460075933322618763952560135426851772191264789... - _Harry J. Smith_, May 16 2009
%t RealDigits[Log[5], 10, 125][[1]] (* _Alonso del Arte_, Oct 04 2014 *)
%o (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
%Y Cf. A016733 (continued fraction). - _Harry J. Smith_, May 16 2009
%K nonn,cons,easy
%O 1,2
%A _N. J. A. Sloane_
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