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Natural Logarithm of 10

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The decimal expansion of the natural logarithm of 10 is given by

ln10=2.302585092994045684...
(1)

(OEIS A002392).

It is also given by the BBP-type formulas

ln10=1/(32)sum_(k=0)^(infty)1/((-64)^k)((64)/(12k+1)+(16)/(12k+2)+8/(12k+4)-(16)/(12k+5)+8/(12k+6)+(12)/(12k+7)-2/(12k+8)+4/(12k+9)+1/(12k+10)-3/(12k+11)-1/(12k+12))
(2)
=1/2sum_(k=0)^(infty)1/((-4)^k)(6/(4k+1)-3/(4k+3)-1/(4k+4))
(3)
=1/(16)sum_(k=0)^(infty)1/(16^k)((24)/(4k+1)+(20)/(4k+2)+6/(4k+3)+1/(4k+4))
(4)
=1/8sum_(k=0)^(infty)1/(16^k)((16)/(8k+1)+8/(8k+2)-8/(8k+3)+4/(8k+4)-4/(8k+5)+2/(8k+6)+2/(8k+7)+1/(8k+8))
(5)
=2/9sum_(k=0)^(infty)1/(81^k)(9/(4k+1)+2/(4k+2)+1/(4k+3))
(6)
=2/(729)sum_(k=0)^(infty)1/(6561^k)((729)/(8k+1)+(162)/(8k+2)+(81)/(8k+3)+9/(8k+5)+2/(8k+6)+1/(8k+7))
(7)

(E. W. Weisstein, Aug. 28, 2008).

See also

BBP-Type Formula, Natural Logarithm, Natural Logarithm of 2, Natural Logarithm of 10 Continued Fraction, Natural Logarithm of 10 Digits

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References

Sloane, N. J. A. Sequence A002392/M0394 in "The On-Line Encyclopedia of Integer Sequences."Mansell, W. E. Tables of Natural and Common Logarithms. Royal Society Mathematical Tables, Vol. 8. Cambridge, England: Cambridge University Press, p. 2, 1964.Uhler, H. S. "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, 205-212, 1940.

Cite this as:

Weisstein, Eric W. "Natural Logarithm of 10." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/NaturalLogarithmof10.html

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