y = ln (kx) / ln (k)

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  • 8/19/2019 y = ln (kx) / ln (k)

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     A^n = 2A

    then n=ln2A / lnA

     A 2A lnA ln2A

    0 0 Err:502 Err:502

    0.1 0.2 -2.302585093 -1.6094379124

    0.2 0.4 -1.6094379124 -0.9162907319

    0.3 0.6 -1.2039728043 -0.5108256238

    0.4 0.8 -0.9162907319 -0.2231435513

    0.5 1 -0.6931471806 0

    0.6 1.2 -0.5108256238 0.1823215568

    0.7 1.4 -0.3566749439 0.3364722366

    0.8 1.6 -0.2231435513 0.4700036292

    0.9 1.8 -0.1053605157 0.5877866649

    1 2 0 0.6931471806 1.1 2.2 0.0953101798 0.7884573604

    1.2 2.4 0.1823215568 0.8754687374

    1.3 2.6 0.2623642645 0.955511445

    1.4 2.8 0.3364722366 1.0296194172

    1.5 3 0.4054651081 1.0986122887

    1.6 3.2 0.4700036292 1.1631508098

    1.7 3.4 0.5306282511 1.2237754316

    1.8 3.6 0.5877866649 1.2809338455

    1.9 3.8 0.6418538862 1.3350010667

    2 4 0.6931471806 1.3862943611

    2.1 4.2 0.7419373447 1.4350845253

    2.2 4.4 0.7884573604 1.4816045409

    2.3 4.6 0.8329091229 1.5260563035

    2.4 4.8 0.8754687374 1.5686159179

    2.5 5 0.9162907319 1.6094379124

    2.6 5.2 0.955511445 1.6486586256

    2.7 5.4 0.993251773 1.6863989536 2.8 5.6 1.0296194172 1.7227665977

    2.9 5.8 1.064710737 1.7578579176

    3 6 1.0986122887 1.7917594692

    3.1 6.2 1.1314021115 1.8245492921

    3.2 6.4 1.1631508098 1.8562979904

    3.3 6.6 1.1939224685 1.887069649

    3.4 6.8 1.2237754316 1.9169226122

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    3.5 7 1.2527629685 1.9459101491

    3.6 7.2 1.2809338455 1.974081026

    3.7 7.4 1.3083328197 2.0014800002

    3.8 7.6 1.3350010667 2.0281482473

    3.9 7.8 1.3609765531 2.0541237337

    4 8 1.3862943611 2.0794415417

    4.1 8.2 1.4109869737 2.1041341543

    4.2 8.4 1.4350845253 2.1282317058

    4.3 8.6 1.4586150227 2.1517622033

    4.4 8.8 1.4816045409 2.1747517215

    4.5 9 1.5040773968 2.1972245773

    4.6 9.2 1.5260563035 2.2192034841

    4.7 9.4 1.5475625087 2.2407096893

    4.8 9.6 1.5686159179 2.2617630985

    4.9 9.8 1.5892352051 2.2823823857

    5 10 1.6094379124 2.302585093

    5.1 10.2 1.6292405397 2.3223877203

    5.2 10.4 1.6486586256 2.3418058061

    5.3 10.6 1.6677068206 2.3608540011

    5.4 10.8 1.6863989536 2.3795461341

    5.5 11 1.7047480922 2.3978952728

    5.6 11.2 1.7227665977 2.4159137783

    5.7 11.4 1.7404661748 2.4336133554

    5.8 11.6 1.7578579176 2.4510050981

    5.9 11.8 1.7749523509 2.4680995315

    6 12 1.7917594692 2.4849066498

    6.1 12.2 1.8082887712 2.5014359517

    6.2 12.4 1.8245492921 2.5176964726

    6.3 12.6 1.8405496334 2.533696814

    6.4 12.8 1.8562979904 2.5494451709

    6.5 13 1.8718021769 2.5649493575 6.6 13.2 1.887069649 2.5802168296

    6.7 13.4 1.9021075264 2.595254707

    6.8 13.6 1.9169226122 2.6100697927

    6.9 13.8 1.9315214116 2.6246685922

    7 14 1.9459101491 2.6390573296

    7.1 14.2 1.960094784 2.6532419646

    7.2 14.4 1.974081026 2.6672282066

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    7.3 14.6 1.9878743482 2.6810215287

    7.4 14.8 2.0014800002 2.6946271808

    7.5 15 2.0149030205 2.7080502011

    7.6 15.2 2.0281482473 2.7212954279

    7.7 15.4 2.0412203289 2.7343675094

    7.8 15.6 2.0541237337 2.7472709143

    7.9 15.8 2.0668627595 2.76000994

    8 16 2.0794415417 2.7725887222

    8.1 16.2 2.0918640617 2.7850112422

    8.2 16.4 2.1041341543 2.7972813348

    8.3 16.6 2.1162555148 2.8094026954

    8.4 16.8 2.1282317058 2.8213788864

    8.5 17 2.1400661635 2.8332133441

    8.6 17.2 2.1517622033 2.8449093838

    8.7 17.4 2.1633230257 2.8564702062

    8.8 17.6 2.1747517215 2.867898902

    8.9 17.8 2.1860512767 2.8791984573

    9 18 2.1972245773 2.8903717579

    9.1 18.2 2.2082744135 2.9014215941

    9.2 18.4 2.2192034841 2.9123506646

    9.3 18.6 2.2300144002 2.9231615807

    9.4 18.8 2.2407096893 2.9338568698

    9.5 19 2.2512917986 2.9444389792

    9.6 19.2 2.2617630985 2.954910279

    9.7 19.4 2.2721258855 2.9652730661

    9.8 19.6 2.2823823857 2.9755295662

    9.9 19.8 2.2925347571 2.9856819377

    10 20 2.302585093 2.9957322736

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    n A^n (=2A)

    0   1

    0.6989700043 0.2

    0.5693234419 0.4

    0.4242833575 0.6

    0.2435292026 0.8

    0 1

    -0.3569154489 1.2

    -0.9433582099 1.4

    -2.1062837195 1.6

    -5.578813479 1.8

    #DIV/0 #DIV/0

    8.2725408973 2.2

    4.8017840169 2.4

    3.6419267958 2.6

    3.0600427171 2.8

    2.7095112914 3

    2.4747698474 3.2

    2.3062764359 3.4

    2.1792495848 3.6

    2.079914285 3.8

    2 4

    1.9342395089 4.2

    1.8791181558 4.4

    1.8322002503 4.6

    1.7917440692 4.8

    1.7564707974 5

    1.7254200713 5.2

    1.6978564745 5.4 1.6732071764 5.6

    1.6510192454 5.8

    1.6309297536 6

    1.6126444113 6.2

    1.5959220204 6.4

    1.5805629753 6.6

    1.5664006342 6.8

    0 2

    -8

    -6

    -4

    -2

    0

    2

    4

    6

    8

    10

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    1.5532947557 7

    1.5411264469 7.2

    1.5297942314 7.4

    1.5192109563 7.6

    1.5093013388 7.8

    1.5 8

    1.4912498793 8.2

    1.4830009441 8.4

    1.4752091332 8.6

    1.4678354861 8.8

    1.4608454206 9

    1.4542081304 9.2

    1.4478960796 9.4

    1.4418845765 9.6

    1.4361514132 9.8

    1.4306765581 10

    1.4254418937 10.2

    1.4204309915 10.4

    1.4156289175 10.6

    1.4110220651 10.8

    1.4065980092 11

    1.4023453795 11.2

    1.3982537498 11.4

    1.3943135413 11.6

    1.3905159371 11.8

    1.3868528072 12

    1.3833166426 12.2

    1.3799004957 12.4

    1.3765979292 12.6

    1.3734029688 12.8

    1.3703100622 13 1.3673140421 13.2

    1.364410093 13.4

    1.3615937212 13.6

    1.3588607283 13.8

    1.3562071871 14

    1.3536294195 14.2

    1.3511239769 14.4

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    1.3486876226 14.6

    1.3463173154 14.8

    1.3440101948 15

    1.3417635676 15.2

    1.3395748958 15.4

    1.3374417856 15.6

    1.3353619767 15.8

    1.3333333333 16

    1.3313538357 16.2

    1.3294215719 16.4

    1.3275347309 16.6

    1.3256915958 16.8

    1.3238905378 17

    1.3221300102 17.2

    1.3204085439 17.4

    1.3187247416 17.6

    1.3170772744 17.8

    1.3154648768 18

    1.3138863433 18.2

    1.3123405247 18.4

    1.3108263249 18.6

    1.3093426979 18.8

    1.3078886446 19

    1.3064632105 19.2

    1.3050654829 19.4

    1.303694589 19.6

    1.3023496932 19.8

    1.3010299957 20

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    4 6

    n vs. A

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    8 10

    n

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     A^n = 2A

    then n=ln2A / lnA

      2A lnA ln2A

    0 0 Err:502 Err:502

    0.1 0.2 -2.302585093 -1.6094379124

    0.2 0.4 -1.6094379124 -0.9162907319

    0.3 0.6 -1.2039728043 -0.5108256238

    0.4 0.8 -0.9162907319 -0.2231435513

    0.5 1 -0.6931471806 0

    0.6 1.2 -0.5108256238 0.1823215568

    0.7 1.4 -0.3566749439 0.3364722366

    0.8 1.6 -0.2231435513 0.4700036292

    0.9 1.8 -0.1053605157 0.5877866649

    1 2 0 0.6931471806

    1.1 2.2 0.0953101798 0.7884573604

    1.2 2.4 0.1823215568 0.8754687374

    1.3 2.6 0.2623642645 0.955511445

    1.4 2.8 0.3364722366 1.0296194172

    1.5 3 0.4054651081 1.0986122887

    1.6 3.2 0.4700036292 1.1631508098

    1.7 3.4 0.5306282511 1.2237754316

    1.8 3.6 0.5877866649 1.2809338455

    1.9 3.8 0.6418538862 1.3350010667

    2 4 0.6931471806 1.3862943611

    2.1 4.2 0.7419373447 1.4350845253

    2.2 4.4 0.7884573604 1.4816045409

    2.3 4.6 0.8329091229 1.5260563035

    2.4 4.8 0.8754687374 1.5686159179

    2.5 5 0.9162907319 1.6094379124

    2.6 5.2 0.955511445 1.6486586256

    2.7 5.4 0.993251773 1.6863989536 2.8 5.6 1.0296194172 1.7227665977

    2.9 5.8 1.064710737 1.7578579176

    3 6 1.0986122887 1.7917594692

    3.1 6.2 1.1314021115 1.8245492921

    3.2 6.4 1.1631508098 1.8562979904

    3.3 6.6 1.1939224685 1.887069649

    3.4 6.8 1.2237754316 1.9169226122

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    3.5 7 1.2527629685 1.9459101491

    3.6 7.2 1.2809338455 1.974081026

    3.7 7.4 1.3083328197 2.0014800002

    3.8 7.6 1.3350010667 2.0281482473

    3.9 7.8 1.3609765531 2.0541237337

    4 8 1.3862943611 2.0794415417

    4.1 8.2 1.4109869737 2.1041341543

    4.2 8.4 1.4350845253 2.1282317058

    4.3 8.6 1.4586150227 2.1517622033

    4.4