OFFSET
1,2
REFERENCES
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 2.
Konrad Knopp, Theory and application of infinite series, Blackie & Son Limited, London and Glasgow, 1954. See p. 198.
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].
Michael I. Shamos, A catalog of the real numbers, (2007). See p. 642.
FORMULA
From Peter Bala, Nov 11 2019: (Start)
log(7) = 2*sqrt(3)*Integral_{t = 0..sqrt(3)/3} (1 - t^4)/(1 + t^6) dt.
log(7) = (8/9)*Sum_{n >= 0} (12*n+11)/((6*n+1)*(6*n+5))*(-1/27)^n.
log(7) = 6*Sum_{n >= 0} ( 243/(12*n+1) - 27/(12*n+5) - 9/(12*n+7) + 1/(12*n+11) )*(1/729)^(n+1), a BPP-type formula. (End)
log(7) = 2*Sum_{n >= 1} 1/(n*P(n, 4/3)*P(n-1, 4/3)), where P(n, x) denotes the n-th Legendre polynomial. The first 20 terms of the series gives the approximation log(7) = 1.945910149055(27...), correct to 12 decimal places. - Peter Bala, Mar 18 2024
From Stefano Spezia, Jan 18 2026: (Start)
log(7) = lim_{n->oo} n*(7^(1/n) - 1) = lim_{k->oo} 2^k*(7^(1/2^k) - 1) [Knopp].
log(7) = Li(1, 6/7) where Li(n, z) is the polylogarithm function [Shamos]. (End)
EXAMPLE
1.945910149055313305105352743443179729637084729581861188459390149937579...
MATHEMATICA
First[RealDigits[Log[7], 10, 100]] (* Paolo Xausa, Mar 21 2024 *)
PROG
(PARI) default(realprecision, 20080); x=log(7); for (n=1, 20000, d=floor(x); x=(x-d)*10; write("b016630.txt", n, " ", d)); \\ Harry J. Smith, May 16 2009
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
STATUS
approved