# Math Help - Find exponential equation given points?

1. ## 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

2. 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

3. Originally Posted by gyan1010
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

4. 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.

5. 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.

6. Originally Posted by gyan1010
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

7. 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.

8. 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 (:

9. I mentioned that i think. I could not find any trend line to match the points for either graph.

10. Originally Posted by gyan1010
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

11. 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 $C \exp( a n^b)$.