# Exponential modeling of G-force

1. Apr 25, 2010

### cgi093

1. The problem statement, all variables and given/known data

Derive an equation of the form y=ax^b to model the given data: (35, 0.01) (28, 0.03) (20, 0.1) (15, 0.3) (11, 1) (9,3) (6, 10) (4.5, 30)

2. Relevant equations

Well, I know the answer is y = (7790 +/- 1246)x^(-3.698+/-0.1036) because that's what LoggerPro spits out, but I don't know how to derive it correctly without a program.

3. The attempt at a solution

Among the many, many attempts:

y = Ax^b
b = log(y)-log(A)
****** log(x)

Then I inserted numbers from two different data points then set the equations equal to each other, resulting in:

log(0.1)-log(A) = log(1)-log(A)
*** log(20) ****** *** log(11)

Which, after a step or two, became:

log(20) = log(11)log(0.1) +log(11)
************* log(A)

So, log(A) = log(11)log(0.1)
*********** log (20) - log(11)

Yielding an answer of A ~ -4.01

However, that isn't right, so I didn't even try to solve for b.

Could anyone please help? This is a big assignment, so sorry if I bump this thread a bit until I get help. And please take me through the steps, because I don't want to copy, I want to understand. I just need some help getting there. Thanks.

P.S. Ignore the asterisks. They're there only to get the denominators in the right place.

2. Apr 25, 2010

### zachzach

I would use the method of least squares.

3. Apr 25, 2010

### lanedance

so starting with
$$y = ax^b$$

taking logs
$$log(y) = log(ax^b) = log(x^b) + log(a)= b.log(x) + log(a)$$

so plotting up log(y) vs log(x) should be a graph of a straight line... if you can work out the line of best fit for that straight line, you should be able to cacaluate a & b from them...

4. Apr 25, 2010

### lanedance

if its really experimental data, the data is not perfect, so you can't use a single pair of data points - you should use the whole data set to find the line of best fit...

as zachzach points of least squares is a good idea

5. Apr 25, 2010

### cgi093

@ zach: Thanks, but no idea how to do that. And I posted this in the wrong subforum, because I'm in advanced pre-calc. So no calculus to help me please.

@lane: it's not experimental. That's an interesting idea though. I'll try it. Thanks a lot.

6. Apr 25, 2010

### lanedance

then just plot the points on a log-log graph & draw a straight line of best fit on graph paer & work out the gradient & intercept

7. Apr 25, 2010

### zachzach

8. Apr 25, 2010

### cgi093

Thanks again, but that looks like gibberish to me. I've never used sigma outside of physics, for example. Maybe it wouldn't be that hard to learn, but I did google least squares and none of it really made sense to me.

9. Apr 25, 2010

### lanedance

least squares is just a statistical method to give you the line of best fit through a set of data points

its derived by calculus, but doesn't require any to use it...

10. Apr 25, 2010

### zachzach

Good point.

11. Apr 25, 2010