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Math Help - Find exponential equation given points?

  1. #1
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    Find exponential equation given points?

    First time I'm asking a math question on any forum that's not related to homework, but I got interested in trying to solve a pattern. Not entirely important but it peaked my curiosity.

    It is an exponential pattern where I have values for x = 2 to 100. It is possible the y values are truncated/rounded decimals as this is from a computer game.

    I can plot the values in excel fairly easily, and get a nice looking exponential graph. What method would i use to find the equation used to generate the numbers though?

    Here they are in 2 columns to save space:
    2 : 42 ~~~~~~~~~~~~~~~~~~~ 52 : 2165947
    3 : 104 ~~~~~~~~~~~~~~~~~~ 53 : 2368499
    4 : 280 ~~~~~~~~~~~~~~~~~~ 54 : 2586879
    5 : 364 ~~~~~~~~~~~~~~~~~~ 55 : 2822194
    6 : 612 ~~~~~~~~~~~~~~~~~~ 56 : 3075629
    7 : 976 ~~~~~~~~~~~~~~~~~~ 57 : 3348452
    8 : 1490 ~~~~~~~~~~~~~~~~~ 58 : 3642020
    9 : 2193 ~~~~~~~~~~~~~~~~~ 59 : 3957785
    10 : 3128 ~~~~~~~~~~~~~~~~ 60 : 4297300
    11 : 4341 ~~~~~~~~~~~~~~~~ 61 : 4662227
    12 : 5885 ~~~~~~~~~~~~~~~~ 62 : 5054090
    13 : 7818 ~~~~~~~~~~~~~~~~ 63 : 5474505
    14 : 10202 ~~~~~~~~~~~~~~~ 64 : 5925185
    15 : 13106 ~~~~~~~~~~~~~~~ 65 : 6407945
    16 : 16606 ~~~~~~~~~~~~~~~ 66 : 6924709
    17 : 20780 ~~~~~~~~~~~~~~~ 67 : 7477218
    18 : 25724 ~~~~~~~~~~~~~~~ 68 : 8067299
    19 : 31540 ~~~~~~~~~~~~~~~ 69 : 8696824
    20 : 38331 ~~~~~~~~~~~~~~~ 70 : 9367757
    21 : 46224 ~~~~~~~~~~~~~~~ 71 : 10082146
    22 : 55350 ~~~~~~~~~~~~~~~ 72 : 10842126
    23 : 65845 ~~~~~~~~~~~~~~~ 73 : 11650062
    24 : 77869 ~~~~~~~~~~~~~~~ 74 : 12508554
    25 : 91588 ~~~~~~~~~~~~~~~ 75 : 13420451
    26 : 107175 ~~~~~~~~~~~~~~ 76 : 14388857
    27 : 124828 ~~~~~~~~~~~~~~ 77 : 15417146
    28 : 144750 ~~~~~~~~~~~~~~ 78 : 16508977
    29 : 167160 ~~~~~~~~~~~~~~ 79 : 17668295
    30 : 192295 ~~~~~~~~~~~~~~ 80 : 18899355
    31 : 220406 ~~~~~~~~~~~~~~ 81 : 20206727
    32 : 251753 ~~~~~~~~~~~~~~ 82 : 21595301
    33 : 286627 ~~~~~~~~~~~~~~ 83 : 23070505
    34 : 325326 ~~~~~~~~~~~~~~ 84 : 24638311
    35 : 368180 ~~~~~~~~~~~~~~ 85 : 26305266
    36 : 415530 ~~~~~~~~~~~~~~ 86 : 28078511
    37 : 467752 ~~~~~~~~~~~~~~ 87 : 29965813
    38 : 525232 ~~~~~~~~~~~~~~ 88 : 31975578
    39 : 588394 ~~~~~~~~~~~~~~ 89 : 34116885
    40 : 657675 ~~~~~~~~~~~~~~ 90 : 36399509
    41 : 733546 ~~~~~~~~~~~~~~ 91 : 38833942
    42 : 816499 ~~~~~~~~~~~~~~ 92 : 41431421
    43 : 907062 ~~~~~~~~~~~~~~ 93 : 44203958
    44 : 1005781 ~~~~~~~~~~~~~ 94 : 47164362
    45 : 1113245 ~~~~~~~~~~~~~ 95 : 50326392
    46 : 1230060 ~~~~~~~~~~~~~ 96 : 53704797
    47 : 1356876 ~~~~~~~~~~~~~ 97 : 57315344
    48 : 1494365 ~~~~~~~~~~~~~ 98 : 61174862
    49 : 1643244 ~~~~~~~~~~~~~ 99 : 65301264
    50 : 1804255 ~~~~~~~~~~~~~ 100 : 69713595
    51 : 1978188
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  2. #2
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    I don't know if this will work, but you can try it out

    since the fomula for an exponential formula is

    y = R^x

    where r is any real number and you have an x and a y value, you could porbably sub them in x = 2 y = 42 take log of both side, isolate for x and solve, .. just a hunch seems to easy to be real
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  3. #3
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    Quote Originally Posted by gyan1010 View Post
    First time I'm asking a math question on any forum that's not related to homework, but I got interested in trying to solve a pattern. Not entirely important but it peaked my curiosity.

    It is an exponential pattern where I have values for x = 2 to 100. It is possible the y values are truncated/rounded decimals as this is from a computer game.

    I can plot the values in excel fairly easily, and get a nice looking exponential graph. What method would i use to find the equation used to generate the numbers though?

    Here they are in 2 columns to save space:
    2 : 42 ~~~~~~~~~~~~~~~~~~~ 52 : 2165947
    3 : 104 ~~~~~~~~~~~~~~~~~~ 53 : 2368499
    4 : 280 ~~~~~~~~~~~~~~~~~~ 54 : 2586879
    5 : 364 ~~~~~~~~~~~~~~~~~~ 55 : 2822194
    6 : 612 ~~~~~~~~~~~~~~~~~~ 56 : 3075629
    7 : 976 ~~~~~~~~~~~~~~~~~~ 57 : 3348452
    8 : 1490 ~~~~~~~~~~~~~~~~~ 58 : 3642020
    9 : 2193 ~~~~~~~~~~~~~~~~~ 59 : 3957785
    10 : 3128 ~~~~~~~~~~~~~~~~ 60 : 4297300
    11 : 4341 ~~~~~~~~~~~~~~~~ 61 : 4662227
    12 : 5885 ~~~~~~~~~~~~~~~~ 62 : 5054090
    13 : 7818 ~~~~~~~~~~~~~~~~ 63 : 5474505
    14 : 10202 ~~~~~~~~~~~~~~~ 64 : 5925185
    15 : 13106 ~~~~~~~~~~~~~~~ 65 : 6407945
    16 : 16606 ~~~~~~~~~~~~~~~ 66 : 6924709
    17 : 20780 ~~~~~~~~~~~~~~~ 67 : 7477218
    18 : 25724 ~~~~~~~~~~~~~~~ 68 : 8067299
    19 : 31540 ~~~~~~~~~~~~~~~ 69 : 8696824
    20 : 38331 ~~~~~~~~~~~~~~~ 70 : 9367757
    21 : 46224 ~~~~~~~~~~~~~~~ 71 : 10082146
    22 : 55350 ~~~~~~~~~~~~~~~ 72 : 10842126
    23 : 65845 ~~~~~~~~~~~~~~~ 73 : 11650062
    24 : 77869 ~~~~~~~~~~~~~~~ 74 : 12508554
    25 : 91588 ~~~~~~~~~~~~~~~ 75 : 13420451
    26 : 107175 ~~~~~~~~~~~~~~ 76 : 14388857
    27 : 124828 ~~~~~~~~~~~~~~ 77 : 15417146
    28 : 144750 ~~~~~~~~~~~~~~ 78 : 16508977
    29 : 167160 ~~~~~~~~~~~~~~ 79 : 17668295
    30 : 192295 ~~~~~~~~~~~~~~ 80 : 18899355
    31 : 220406 ~~~~~~~~~~~~~~ 81 : 20206727
    32 : 251753 ~~~~~~~~~~~~~~ 82 : 21595301
    33 : 286627 ~~~~~~~~~~~~~~ 83 : 23070505
    34 : 325326 ~~~~~~~~~~~~~~ 84 : 24638311
    35 : 368180 ~~~~~~~~~~~~~~ 85 : 26305266
    36 : 415530 ~~~~~~~~~~~~~~ 86 : 28078511
    37 : 467752 ~~~~~~~~~~~~~~ 87 : 29965813
    38 : 525232 ~~~~~~~~~~~~~~ 88 : 31975578
    39 : 588394 ~~~~~~~~~~~~~~ 89 : 34116885
    40 : 657675 ~~~~~~~~~~~~~~ 90 : 36399509
    41 : 733546 ~~~~~~~~~~~~~~ 91 : 38833942
    42 : 816499 ~~~~~~~~~~~~~~ 92 : 41431421
    43 : 907062 ~~~~~~~~~~~~~~ 93 : 44203958
    44 : 1005781 ~~~~~~~~~~~~~ 94 : 47164362
    45 : 1113245 ~~~~~~~~~~~~~ 95 : 50326392
    46 : 1230060 ~~~~~~~~~~~~~ 96 : 53704797
    47 : 1356876 ~~~~~~~~~~~~~ 97 : 57315344
    48 : 1494365 ~~~~~~~~~~~~~ 98 : 61174862
    49 : 1643244 ~~~~~~~~~~~~~ 99 : 65301264
    50 : 1804255 ~~~~~~~~~~~~~ 100 : 69713595
    51 : 1978188
    Assume that the data approximately satisfies an equation of the form:

    y=e^{\lambda (t+k)}

    Take logs:

    \ln(y)=\lambda t + \lambda k

    The write Y for \ln(y) and K for \lambda k

    Now plot Y against t, if you have a reasonable approximation to a straight line then the slope is \lambda and the intercept is K.


    CB
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  4. #4
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    ok i took t as the 1-100 and Y as natural log of the larger and larger numbers.

    I didn't get a strait line. Does that mean there are other unknown variables?

    Here are my excel graphs, of both (t , y) and (t , ln(y)). I was unable to plot a trendline in excel to exactly match either of them.

    Attached Thumbnails Attached Thumbnails Find exponential equation given points?-1.png   Find exponential equation given points?-2.png  
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  5. #5
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    Could you please explain the latter half of that?

    I got to the natural log of y, but I couldn't follow you after that.
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  6. #6
    Grand Panjandrum
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    Quote Originally Posted by gyan1010 View Post
    ok i took t as the 1-100 and Y as natural log of the larger and larger numbers.

    I didn't get a strait line. Does that mean there are other unknown variables?

    Here are my excel graphs, of both (t , y) and (t , ln(y)). I was unable to plot a trendline in excel to exactly match either of them.
    Well the plots of the data looks like y=e^{kt^{1/2}} may be an acceptable representation. Now take logs to get:

    Y=\ln(y)=\lambda t^{1/2}

    or:

    Y^2=\lambda^2 t

    So now try plotting Y^2 against t, and if this is a line through the origin the slope will be \lambda^2

    If this does not give you a straight line then you do not have a simple exponential model and rather more work would be needed to find a good representation.

    CB
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  7. #7
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    Actually after posting that last one I did try that ln(t) squared which straitened it out a little. Then cubed even more so but it was varying above and below the strait line at that point. More of a wavy strait line.

    My original theory was that it was some exponential that had a component of the previous result. A function with memory is the term i use in my signals & system course. I just don't really know how to test that.

    This is all for fun anyway though. The values are actually levels in a video game and large number is the exact experience required to reach that level. I left out 1 because you start there. Figured i would use math to try and fulfill my curiosity about the equation the developers used.
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  8. #8
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    since u did plot the points on excel,

    there is an option on excel if u click the points then right click, and pick add a tread line then set it to exponential and say display equation on chart, you should get the equation (:
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  9. #9
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    I mentioned that i think. I could not find any trend line to match the points for either graph.
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  10. #10
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    Quote Originally Posted by gyan1010 View Post
    Actually after posting that last one I did try that ln(t) squared which straitened it out a little. Then cubed even more so but it was varying above and below the strait line at that point. More of a wavy strait line.

    My original theory was that it was some exponential that had a component of the previous result. A function with memory is the term i use in my signals & system course. I just don't really know how to test that.

    This is all for fun anyway though. The values are actually levels in a video game and large number is the exact experience required to reach that level. I left out 1 because you start there. Figured i would use math to try and fulfill my curiosity about the equation the developers used.
    It might be usefull if you could post the data in a form that can be dumped into Excel without retyping so that others can play with this.

    Given its from a game, and the apparent smoothness of the curves there is probably some relativly simple relation being used to generate the data.

    CB
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  11. #11
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    Plotting "log log f(n)" against "log n", you get approximately a line.
    I found the plot coincides pretty much with \exp(4 n^{0.327}), but It could be possible that for instance the original data is in fact of the form C \exp( a n^b).
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