An Example of Exponential Decay and Logarithmic Plotting

Gregg Geist
Apple Learning Interchange

The following description is intended to illustrate for a teacher of secondary science or mathematics the complete process of observing an exponential decay phenomenon and calculating the resulting exponential function. The amplitude of an exponentially decaying variable is measured, as well as the change in amplitude between time increments. The results are tabulated along with the natural logarithms of the amplitudes and changes in amplitude. These are used to produce an exponential decay curve, log plots of the same data (which are theoretically linear) and an exponential fit to the data. It is also shown that the decay constant for the amplitude is the same as that for the change in amplitude, since change in amplitude = (constant) x (amplitude). The description below is quite detailed, and all of that detail would only be usable by pre-calculus students and higher; however, the experiment, creation of much of the table, and plotting of the exponential decay curve in order to understand its shape would be accessable even to pre-algebra level students and certainly to students with a background in algebra. Error values are included primarily to give an idea of their magnitude to teachers, and should not be included in a first treatment of exponential functions, since they probably would distract from the learning about exponential functions themselves. Students should always be aware, however, that uncontrollable factors always randomize data or measurements to some extent, and that the analysis of graphs of these data to find slopes and intercepts of lines will usually yield results that disagree from theoretical values.

For more examples of exponential growth and decay processes (without analyses) go to Exponential Growth and Decay Experiments.

Proceedure and Hypothesis

In the experiment, the rolling of dice was used to produce an example exponential decay. 165 dice were rolled, and the dice that showed a 6 face up were removed, with the number removed being written down so that the number remaining could be calculated. This process continued until there were no dice remaining.

Theoretically, the number of dice in any turn should be 5/6 times the number in the previous turn. This produces the following function for N:
           N = 165(⁵/₆)^t; where t is the number of turns.
To write this in terms of the natural base e we need to find a decay constant k for which
           (⁵/₆)^t = e^kt
Taking the natural logarithm of both sides and using some simplification rules gives:
           tln(⁵/₆) = kt
         k = ln((⁵/₆) = -0.1823 (approximately)
Therefore, the exponential curve produced by this process should be:
           N = 165e^-0.1823t

The following table of the actual results was created, using an Excel spreadsheet. N is the number of dice and DeltaN is the number "lost" each turn. The fraction DeltaN/N was also calculated, which should be close to 1/6, and the natural logarithm of each of N and DeltaN were also calculated.

Turn	N	DeltaN	fraction	ln(N)	ln(DeltaN)
0	165	24	0.145	5.106	3.178
1	141	24	0.170	4.949	3.178
2	117	20	0.171	4.762	2.996
3	97	10	0.103	4.575	2.303
4	87	17	0.195	4.466	2.833
5	70	9	0.129	4.249	2.198
6	61	10	0.164	4.119	2.303
7	51	8	0.157	3.932	2.079
8	43	5	0.1167	3.761	1.609
9	38	4	0.1057	3.638	1.386
10	34	2	0.059	3.526	0.693
11	32	10	0.313	3.466	2.303
12	22	1	0.046	3.091	0
13	21	5	0.238	3.045	1.609
14	16	4	0.250	2.773	1.386
15	12	2	0.167	2.485	0.693
16	10	2	0.200	2.303	0.693
17	8	0	0	2.079	error
18	8	1	0.125	2.079	0
19	7	3	0.429	1.946	1.099
20	4	0	0	1.386	error
21	4	1	0.250	1.386	0
22	3	1	0.333	1.099	0
23	2	1	0.500	0.693	0
24	1	0	0	0	error
25	1	0	0	0	error
26	1	0	0	0	error
27	1	0	0	0	error
28	1	0	0	0	error
29	1	1	1	0	0
30	0	0	error	error	error

Data Analysis

The data in the table were used to generate the following plots. First, the plot of N vs. t (columns 1 and 2 of the table) which shows the classical exponential decay curve:

To get a decay constant k for this curve, we plot the natural log of N vs. t instead, since the logarithm converts geometric progressions into linear ones. This generates the following plot, using columns 1 and 5 of the table:

This is linear at the beginning, but then becomes scattered later. The scatter is a phenomenon of small number statistics. Any casino owner knows that on the whole, large numbers of die rolls follow certain rules, but for small numbers, exact results cannot be predicted. The truly sporadic results begin to occur for ln(Dice Remaining) = 2, which corresponds to around 7 dice - a small number for making predictions. The red line is an eyeball best fit to the first ten points. The coordinates of the endpoints of that line segment (shown) give a slope of -0.17ą0.01 and intercept 5.10ą0.01. The equation of the line is therefore:
lnN = 5.10ą0.01 - (0.17ą0.01)t
The ą error values are based on estimates of how much the slope and intercept could be varied without the resulting line being obviously wrong.

Now we convert back to N from lnN, undoing the transformation that produced column 5 and reproducing the exponential function:
         N = e^5.10ą0.01e^{-(0.17ą0.01)t}
       N = (164ą2)e^{-(0.17ą0.01)t}
Which is quite close to the theoretical value of:
         N = 165e^0.1823t

The rate of change of the number of dice should have the same decay constant, since the number of dice removed, theoretically, is always 1/6 the total number rolled. To see if this is true in our experiment we repeat the process of calculating the function, this time for the change in the number of dice. The logarithmic plot is:

This plot is considerably more scattered, since the number of dice removed was always small, statistically speaking, since we started with only 165 dice. Nevertheless, there are a few "nearly" collinear points at the beginning which were used to as a basis for the red line. The equation of that line is:
           lnN = 2.2ą0.1 - (0.18ą0.02)t
Note that the error values are much larger. Proceeding as before, we get for the equation of the exponential decay curve:
           N = (25ą2)e^{-(0.18ą0.02)t}
Of course, actually the change in the number of dice is always negative. Taking the logarithm of a bunch of negative numbers would have really confused the spreadsheet program, so we couldn't do it directly. (If you know complex numbers well, you can do the entire process exactly as above with the negative numbers and come directly to the final result.) In fact, the equation for the change in the number of dice is the negative of the previous equation, or:
           N = -(25ą2)e^{-(0.18ą0.02)t}
Theoretically, the values for DeltaN should be (¹/₆)N if these are fair dice; therefore, using the theoretical equation for N, DeltaN/turn should be given by:
           DeltaN/turn = -27.5e^-0.1823t
In this case, the measured decay constant is very close to the theoretical value but the initial value is at the edge of the range.

Results and Questions

As can be seen, it was possible to simply generate data, tabulate it, create an exponential decay curve, and match that data to exponential functions which were close to theoretical ones for the methods used to produce it.

Students who generated equations to fit their data should be also be able to answer the following questions:

           What would the theoretical function be if 10,000 pennies were tossed, and all heads were removed each turn?
           What would it be if 500 twelve-sided gaming dice were used, and all twelves were removed? What if ones and twos were removed?
           If you started with $100.00 and at the end of every day increased your wealth by 0.1%, what exponential function would give your resulting wealth on any day thereafter? How wealthy would you be in fifty years?

For more examples of experiments which theoretically produce exponential growth or decay curves, go to Exponential Growth and Decay Experiments.