Proof that something decay exponentially?

For a piece of coursework I tried to see if beer foam decayed exponentially. If I looked at my results the height decreased quickly in the first few seconds and then slowed down. Looking at these results it seemed as though it was decaying exponentially, but I can't exactly say "It looks exponential therefore it is".

I asked my teacher for help, and he told me that if I take ln of my answers, plot them and get a straight line it proves that it is exponential. But after handing in my work another teacher told me that this was wrong.

Now I have to do a presentation, and the teacher that marked my work says I need to prove it's exponential or I won't get a good mark.

tl;dr, got some results from an experiment, how do I prove they decrease exponentially with time?

Reply 1

SimonM

The first teacher is correct. Plotting log of height against time yielding a straight line is evidence for your conclusion

Reply 2

kashim91

Original post by sarapon

There should be a common ratio between successive results. So the 20th result divide by the 19th result, = a constant. But to reduce perecentage error, I would rather use 20th result /10th result and the 30th result /40 results, as this would increase the accuracy. I guess your teacher is not completely satisfied with the graph method as this would work on a non exponential function too. If you had y=7X^2 then you could do ln of that and get a straight line graph. You would get
lny=ln7+2lnx

Reply 3

Jtking3000

should be right, plotting the ln of the height against time should give a straight line with negative gradient and y intercept of natural log of initial height

Reply 4

SimonM

Original post by kashim91

Actually, it wouldn't. If you plot log of height against log of time and you obtain a straight line then it satisfies a power law:

y = Ax^n

. But her method is different.

Plotting, log of height against time and you get a straight line, then they are related exponentially.

Reply 5

Charries

Takings log and observing a straight line should suffice, but perhaps the second teacher wants you to find the values of the intercept and the gradient to use in y=-ke^at, and plot that against your data. He might also be thinking that the straight line fit, which presumeably doesn't go right through each of your points, has some error. You could try to find the mathematical equation of the regression line (see excel), or do it yourself (if you've done statistics), to find the value of r, which indicates how close a fit your data is to a straight line.

Reply 6

Jake200493

The first teacher was right because that's what I had to do and I got an A in that piece of coursework, so it must be right xD

Reply 7

sarapon

So if I were to say about the straight line again during my presentation is it likely I'd actually get marked down?

Reply 8

Jake200493

No, I wouldn't say that. I'd say that mentioning it as the proof is the right answer, so that's just what you gotta write.

Reply 9

Cora Lindsay

None of these looks like proof. All you are doing is fitting a function to data. For example, a slow exponential decay measured over a relatively short time interval fits a linear function pretty well. Then there is the uncertainty in your data. Your points are not absolutes, so you may be able to get several different functions to pass through the error bars.

You can do statistical tests to show that the exponential fits your data better than other functions, which is circumstantial evidence (and may be good enough), but not proof. Proof would require you to develop a physical description of the process and show that you'd expect an exponential decay.