Natural Logarithm of 10

Author

Eric W. Weisstein
August 31, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/Constants/NaturalLogarithmof10.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/NaturalLogarithmof10.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Value

N[Log[10],20]

Sums

ln(10)==(1/8192)(∑_(k=0)^∞)(1/(((-16384))^k)) ((24576/(28 k+1))-(12288/(28 k+3))-(4096/(28 k+4))-(6144/(28 k+5))+(3072/(28 k+7))+(1024/(28 k+8))+(1536/(28 k+9))-(768/(28 k+11))-(256/(28 k+12))-(384/(28 k+13))+(192/(28 k+15))+(64/(28 k+16))+(96/(28 k+17))-(48/(28 k+19))-(16/(28 k+20))-(24/(28 k+21))+(12/(28 k+23))+(4/(28 k+24))+(6/(28 k+25))-(3/(28 k+27))-(1/(28 k+28)))

(1/8192)*Sum[(24576/(1 + 28*k) - 12288/(3 + 28*k) - 4096/(4 + 28*k) - 
       6144/(5 + 28*k) + 3072/(7 + 28*k) + 1024/(8 + 28*k) + 1536/(9 + 28*k) - 768/(11 + 28*k) - 
       256/(12 + 28*k) - 384/(13 + 28*k) + 192/(15 + 28*k) + 64/(16 + 28*k) + 96/(17 + 28*k) - 
       48/(19 + 28*k) - 16/(20 + 28*k) - 24/(21 + 28*k) + 12/(23 + 28*k) + 4/(24 + 28*k) + 
       6/(25 + 28*k) - 3/(27 + 28*k) - 1/(28 + 28*k))/(-16384)^k, {k, 0, Infinity}]

FullSimplify[%]

FullSimplify[FunctionExpand[%]]

ln(10)==(1/512) (∑_(k=0)^∞)(1/(((-1024))^k)) ((1024/(20 k+1))+(768/(20 k+2))-(768/(20 k+3))+(384/(20 k+4))-(256/(20 k+5))-(192/(20 k+6))+(192/(20 k+7))-(96/(20 k+8))+(64/(20 k+9))-(32/(20 k+10))-(48/(20 k+11))+(24/(20 k+12))-(16/(20 k+13))-(12/(20 k+14))-(8/(20 k+15))-(6/(20 k+16))+(4/(20 k+17))+(3/(20 k+18))-(3/(20 k+19))-(1/(20 k+20)))

(1/512)*Sum[(1024/(1 + 20*k) + 768/(2 + 20*k) - 768/(3 + 20*k) + 384/(4 + 20*k) - 
       256/(5 + 20*k) - 192/(6 + 20*k) + 192/(7 + 20*k) - 96/(8 + 20*k) + 64/(9 + 20*k) - 
       32/(10 + 20*k) - 48/(11 + 20*k) + 24/(12 + 20*k) - 16/(13 + 20*k) - 12/(14 + 20*k) - 
       8/(15 + 20*k) - 6/(16 + 20*k) + 4/(17 + 20*k) + 3/(18 + 20*k) - 3/(19 + 20*k) - 1/(20 + 20*k))/
      (-1024)^k, {k, 0, Infinity}]

FullSimplify[%]

FullSimplify[FunctionExpand[%]]

ln(10)==(1/32) (∑_(k=0)^∞)(1/(((-64))^k)) ((64/(12 k+1))+(16/(12 k+2))+(8/(12 k+4))-(16/(12 k+5))+(8/(12 k+6))+(12/(12 k+7))-(2/(12 k+8))+(4/(12 k+9))+(1/(12 k+10))-(3/(12 k+11))-(1/(12 k+12)))

Identity[(1/32)*Sum[(64/(1 + 12*k) + 16/(2 + 12*k) + 8/(4 + 12*k) - 
      16/(5 + 12*k) + 8/(6 + 12*k) + 12/(7 + 12*k) - 2/(8 + 12*k) + 
      4/(9 + 12*k) + 1/(10 + 12*k) - 3/(11 + 12*k) - 1/(12 + 12*k))/
     (-64)^k, {k, 0, Infinity}]]

FullSimplify[%]

FunctionExpand[%]//FullSimplify

N[%]

Log[10.]

ln(10)==(1/2) (∑_(k=0)^∞)(1/(((-4))^k)) ((6/(4 k+1))-(3/(4 k+3))-(1/(4 k+4)))

(1/2) (∑_(k=0)^∞)(((-4))^(-k)) ((6/(1+4 k))-(3/(3+4 k))-(1/(4+4 k)))//ReleaseHold

FullSimplify[%]

(1/16) (∑_(k=0)^∞)(1/(16^k))((24/(4 k+1))+(20/(4 k+2))+(6/(4 k+3))+(1/(4 k+4)))==ln(10)

(1/16) Sum[(1/(16^k))((24/(1+4 k))+(20/(2+4 k))+(6/(3+4 k))+(1/(4+4 k))),{k,0,∞}]

FullSimplify[%]

(1/8)(∑_(k=0)^∞)(1/(16^k))((16/(8 k+1))+(8/(8 k+2))-(8/(8 k+3))+(4/(8 k+4))-(4/(8 k+5))+(2/(8 k+6))+(2/(8 k+7))+(1/(8 k+8)))==ln(10)

(1/8)Sum[(1/(16^k))((16/(1+8 k))+(8/(2+8 k))-(8/(3+8 k))+(4/(4+8 k))-(4/(5+8 k))+(2/(6+8 k))+(2/(7+8 k))+(1/(8+8 k))),{k,0,∞}]

FullSimplify[%]

(2/9) (∑_(k=0)^∞)(1/(81^k))((9/(4 k+1))+(2/(4 k+2))+(1/(4 k+3)))==ln(10)

(2/9) Sum[(1/(81^k))((9/(1+4 k))+(2/(2+4 k))+(1/(3+4 k))),{k,0,∞}]

FullSimplify[%]

ln(10)==(1/256) (∑_(k=0)^∞)(1/(256^k)) ((384/(8 k+1))+(320/(8 k+2))+(96/(8 k+3))+(16/(8 k+4))+(24/(8 k+5))+(20/(8 k+6))+(6/(8 k+7))+(1/(8 k+8)))

(1/256)*Sum[(384/(1 + 8*k) + 320/(2 + 8*k) + 96/(3 + 8*k) + 16/(4 + 8*k) + 
       24/(5 + 8*k) + 20/(6 + 8*k) + 6/(7 + 8*k) + 1/(8 + 8*k))/256^k, {k, 0, Infinity}]

FullSimplify[%]

ln(10)==(1/128) (∑_(k=0)^∞)(1/(256^k)) ((256/(16 k+1))+(128/(16 k+2))-(128/(16 k+3))+(64/(16 k+4))-(64/(16 k+5))+(32/(16 k+6))+(32/(16 k+7))+(16/(16 k+8))+(16/(16 k+9))+(8/(16 k+10))-(8/(16 k+11))+(4/(16 k+12))-(4/(16 k+13))+(2/(16 k+14))+(2/(16 k+15))+(1/(16 k+16)))

(1/128)*Sum[(256/(1 + 16*k) + 128/(2 + 16*k) - 128/(3 + 16*k) + 64/(4 + 16*k) - 
       64/(5 + 16*k) + 32/(6 + 16*k) + 32/(7 + 16*k) + 16/(8 + 16*k) + 16/(9 + 16*k) + 8/(10 + 16*k) - 
       8/(11 + 16*k) + 4/(12 + 16*k) - 4/(13 + 16*k) + 2/(14 + 16*k) + 2/(15 + 16*k) + 1/(16 + 16*k))/
      256^k, {k, 0, Infinity}]

FullSimplify[%]

FullSimplify[FunctionExpand[%]]

ln(10)==(1/4096)(∑_(k=0)^∞)(1/(4096^k)) ((6144/(12 k+1))+(5120/(12 k+2))+(1536/(12 k+3))+(256/(12 k+4))+(384/(12 k+5))+(320/(12 k+6))+(96/(12 k+7))+(16/(12 k+8))+(24/(12 k+9))+(20/(12 k+10))+(6/(12 k+11))+(1/(12 k+12)))

Sum[(6144/(1 + 12*k) + 5120/(2 + 12*k) + 1536/(3 + 12*k) + 256/(4 + 12*k) + 
       384/(5 + 12*k) + 320/(6 + 12*k) + 96/(7 + 12*k) + 16/(8 + 12*k) + 24/(9 + 12*k) + 
       20/(10 + 12*k) + 6/(11 + 12*k) + 1/(12 + 12*k))/4096^k, {k, 0, Infinity}]/4096

FullSimplify[%]

ln(10)==(1/2048)(∑_(k=0)^∞)(1/(4096^k)) ((3072/(24 k+1))+(2048/(24 k+2))+(512/(24 k+3))+(512/(24 k+4))+(256/(24 k+5))+(512/(24 k+6))+(512/(24 k+7))+(256/(24 k+8))+(64/(24 k+9))+(640/(24 k+10))+(192/(24 k+12))-(48/(24 k+13))+(16/(24 k+14))-(8/(24 k+15))-(80/(24 k+16))-(4/(24 k+17))-(8/(24 k+19))+(10/(24 k+20))-(1/(24 k+21))+(9/(24 k+22))+(1/(24 k+24)))

(1/2048)*Sum[(3072/(1 + 24*k) + 2048/(2 + 24*k) + 512/(3 + 24*k) + 512/(4 + 24*k) + 
       256/(5 + 24*k) + 512/(6 + 24*k) + 512/(7 + 24*k) + 256/(8 + 24*k) + 64/(9 + 24*k) + 
       640/(10 + 24*k) + 192/(12 + 24*k) - 48/(13 + 24*k) + 16/(14 + 24*k) - 8/(15 + 24*k) - 
       80/(16 + 24*k) - 4/(17 + 24*k) - 8/(19 + 24*k) + 10/(20 + 24*k) - 1/(21 + 24*k) + 9/(22 + 24*k) + 
       1/(24 + 24*k))/4096^k, {k, 0, Infinity}]

FullSimplify[%]

FullSimplify[FunctionExpand[%]]

ln(10)==(2/729) (∑_(k=0)^∞)(1/(6561^k)) ((729/(8 k+1))+(162/(8 k+2))+(81/(8 k+3))+(9/(8 k+5))+(2/(8 k+6))+(1/(8 k+7)))

(2/729) (∑_(k=0)^∞)(6561^(-k)) ((729/(1+8 k))+(162/(2+8 k))+(81/(3+8 k))+(9/(5+8 k))+(2/(6+8 k))+(1/(7+8 k)))//ReleaseHold

FullSimplify[%]

ln(10)==(1/65536)(∑_(k=0)^∞)(1/(65536^k)) ((98304/(16 k+1))+(81920/(16 k+2))+(24576/(16 k+3))+(4096/(16 k+4))+(6144/(16 k+5))+(5120/(16 k+6))+(1536/(16 k+7))+(256/(16 k+8))+(384/(16 k+9))+(320/(16 k+10))+(96/(16 k+11))+(16/(16 k+12))+(24/(16 k+13))+(20/(16 k+14))+(6/(16 k+15))+(1/(16 k+16)))

(1/65536)*Sum[(98304/(1 + 16*k) + 81920/(2 + 16*k) + 24576/(3 + 16*k) + 
       4096/(4 + 16*k) + 6144/(5 + 16*k) + 5120/(6 + 16*k) + 1536/(7 + 16*k) + 256/(8 + 16*k) + 
       384/(9 + 16*k) + 320/(10 + 16*k) + 96/(11 + 16*k) + 16/(12 + 16*k) + 24/(13 + 16*k) + 
       20/(14 + 16*k) + 6/(15 + 16*k) + 1/(16 + 16*k))/65536^k, {k, 0, Infinity}]

FullSimplify[%]

FullSimplify[FunctionExpand[%]]

ln(10)==(1/32768)(∑_(k=0)^∞)(1/(65536^k)) ((65536/(32 k+1))+(32768/(32 k+2))-(32768/(32 k+3))+(16384/(32 k+4))-(16384/(32 k+5))+(8192/(32 k+6))+(8192/(32 k+7))+(4096/(32 k+8))+(4096/(32 k+9))+(2048/(32 k+10))-(2048/(32 k+11))+(1024/(32 k+12))-(1024/(32 k+13))+(512/(32 k+14))+(512/(32 k+15))+(256/(32 k+16))+(256/(32 k+17))+(128/(32 k+18))-(128/(32 k+19))+(64/(32 k+20))-(64/(32 k+21))+(32/(32 k+22))+(32/(32 k+23))+(16/(32 k+24))+(16/(32 k+25))+(8/(32 k+26))-(8/(32 k+27))+(4/(32 k+28))-(4/(32 k+29))+(2/(32 k+30))+(2/(32 k+31))+(1/(32 k+32)))

(1/32768)*Sum[(65536/(1 + 32*k) + 32768/(2 + 32*k) - 32768/(3 + 32*k) + 
       16384/(4 + 32*k) - 16384/(5 + 32*k) + 8192/(6 + 32*k) + 8192/(7 + 32*k) + 4096/(8 + 32*k) + 
       4096/(9 + 32*k) + 2048/(10 + 32*k) - 2048/(11 + 32*k) + 1024/(12 + 32*k) - 1024/(13 + 32*k) + 
       512/(14 + 32*k) + 512/(15 + 32*k) + 256/(16 + 32*k) + 256/(17 + 32*k) + 128/(18 + 32*k) - 
       128/(19 + 32*k) + 64/(20 + 32*k) - 64/(21 + 32*k) + 32/(22 + 32*k) + 32/(23 + 32*k) + 
       16/(24 + 32*k) + 16/(25 + 32*k) + 8/(26 + 32*k) - 8/(27 + 32*k) + 4/(28 + 32*k) - 4/(29 + 32*k) + 
       2/(30 + 32*k) + 2/(31 + 32*k) + 1/(32 + 32*k))/65536^k, {k, 0, Infinity}]

FullSimplify[%]

FullSimplify[FunctionExpand[%]]

ln(10)==((2 )/59049)(∑_(k=0)^∞)(1/(531441^k)) ((59049/(12 k+1))+(13122/(12 k+2))+(6561/(12 k+3))+(729/(12 k+5))+(162/(12 k+6))+(81/(12 k+7))+(9/(12 k+9))+(2/(12 k+10))+(1/(12 k+11)))

BasisSum[{Log[10],{-118098,-26244,-13122,0,-1458,-324,-162,0,-18,-4,-2,0,59049},531441,12,1},k]

(2*Sum[(59049/(1 + 12*k) + 13122/(2 + 12*k) + 6561/(3 + 12*k) + 729/(5 + 12*k) + 
        162/(6 + 12*k) + 81/(7 + 12*k) + 9/(9 + 12*k) + 2/(10 + 12*k) + 1/(11 + 12*k))/531441^k, 
      {k, 0, Infinity}])/59049

FullSimplify[%]

ln(10)==(1/59049)(∑_(k=0)^∞)(1/(531441^k)) ((177147/(24 k+1))-(59049/(24 k+2))-(6561/(24 k+4))-(19683/(24 k+5))-(26244/(24 k+6))-(6561/(24 k+7))-(6561/(24 k+8))-(729/(24 k+10))+(729/(24 k+11))+(648/(24 k+12))+(243/(24 k+13))-(81/(24 k+14))-(81/(24 k+16))-(27/(24 k+17))-(36/(24 k+18))-(9/(24 k+19))-(1/(24 k+20))-(1/(24 k+22))+(1/(24 k+23)))

(1/59049)*Sum[(177147/(1 + 24*k) - 59049/(2 + 24*k) - 6561/(4 + 24*k) - 
       19683/(5 + 24*k) - 26244/(6 + 24*k) - 6561/(7 + 24*k) - 6561/(8 + 24*k) - 729/(10 + 24*k) + 
       729/(11 + 24*k) + 648/(12 + 24*k) + 243/(13 + 24*k) - 81/(14 + 24*k) - 81/(16 + 24*k) - 
       27/(17 + 24*k) - 36/(18 + 24*k) - 9/(19 + 24*k) - 1/(20 + 24*k) - 1/(22 + 24*k) + 1/(23 + 24*k))/
      531441^k, {k, 0, Infinity}]

FullSimplify[%]

FunctionExpand[%]//FullSimplify

ln(10)==(1/59049)(∑_(k=0)^∞)(1/(531441^k)) ((177147/(24 k+2))+(118098/(24 k+3))+(52488/(24 k+4))-(19683/(24 k+5))-(6561/(24 k+8))-(4374/(24 k+9))+(729/(24 k+11))+(648/(24 k+12))+(243/(24 k+14))+(162/(24 k+15))-(27/(24 k+17))-(1/(24 k+20))-(6/(24 k+21))+(1/(24 k+23)))

(1/59049)*Sum[(177147/(2 + 24*k) + 118098/(3 + 24*k) + 52488/(4 + 24*k) - 
       19683/(5 + 24*k) - 6561/(8 + 24*k) - 4374/(9 + 24*k) + 729/(11 + 24*k) + 648/(12 + 24*k) + 
       243/(14 + 24*k) + 162/(15 + 24*k) - 27/(17 + 24*k) - 1/(20 + 24*k) - 6/(21 + 24*k) + 
       1/(23 + 24*k))/531441^k, {k, 0, Infinity}]

FullSimplify[%]

FunctionExpand[%]//FullSimplify

ln(10)==(2/4782969) (∑_(k=0)^∞)(1/(43046721^k)) ((4782969/(16 k+1))+(1062882/(16 k+2))+(531441/(16 k+3))+(59049/(16 k+5))+(13122/(16 k+6))+(6561/(16 k+7))+(729/(16 k+9))+(162/(16 k+10))+(81/(16 k+11))+(9/(16 k+13))+(2/(16 k+14))+(1/(16 k+15)))

(2*Sum[(4782969/(1 + 16*k) + 1062882/(2 + 16*k) + 531441/(3 + 16*k) + 
        59049/(5 + 16*k) + 13122/(6 + 16*k) + 6561/(7 + 16*k) + 729/(9 + 16*k) + 162/(10 + 16*k) + 
        81/(11 + 16*k) + 9/(13 + 16*k) + 2/(14 + 16*k) + 1/(15 + 16*k))/43046721^k, {k, 0, Infinity}])/
    4782969

FullSimplify[%]

FunctionExpand[%]//FullSimplify

BBP-Type Formula

Search

<<Utilities`Simplify`

Monitor[Do[If[(b=FindBasis[Log[10],-6^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,1}],{-6^k,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Log[10],-5^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,1}],{-5^k,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Log[10],-3^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,1}],{-3^k,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Log[10],-3^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,2,2}],{-3^k,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Log[10],-3^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,3,3}],{-3^k,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Log[10],-2^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,1}],{-2^k,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Log[10],-2^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,2,2}],{-2^k,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Log[10],-2^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,3,3}],{-2^k,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Log[10],-1,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,0,0},{n,64},{p,1}],{-1,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Log[10],-1,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,0,0},{n,64},{p,2,2}],{-1,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Log[10],-1,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,0,0},{n,64},{p,3,3}],{-1,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Log[10],1,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,0,0},{n,64},{p,2,2}],{1,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Log[10],1,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,0,0},{n,64},{p,3,3}],{1,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Log[10],2^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,1}],{2^k,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Log[10],2^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,2,2}],{2^k,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Log[10],2^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,3,3}],{2^k,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Log[10],3^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,1}],{3^k,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Log[10],3^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,2,2}],{3^k,n,p}];Print["DONE"]

Monitor[Do[If[(b=FindBasis[Log[10],3^k,n,BasisPower->p,WorkingPrecision->300])=!={},Print[b]],{k,16},{n,64},{p,3,3}],{3^k,n,p}];Print["DONE"]