Question

How do you Find the exponential growth function that models a given data set?

Answer 1

This is only a very simplified version!

Explanation:

Ok, you can use a program such as Excel that has a function to evaluate the fitting line to a set of data but you can do it by yourself using a general function like:
`y=Ae^(kx)`
Where `A` and `k` are two constants you need to evaluate.
To evaluate the constants you use your data.
Say that you have values for `x` and corresponding values for `y`; you substitute them into your basic function and evaluate the constants.

For example:
Consider two pairs of data (I invented them):
`x=0` , `y=5`
And
`x=10` , `y=100`
We use them into our basic function to get:
`5=Ae^(0x)` where `e^0=1`
So we get `A=5`

And

`100=Ae^(10k)` but `A=5` so:
`100=5e^(10k)`
Rearranging:
`e^(10k)=100/5`
`e^(10k)=20`
Take the natural log of both sides to get rid of the exponential:
`10k=ln(20)`
`k=ln(20)/10=0.3`

So finally, the exponential function modelling our data will be:

`y=5e^(0.3x)`

Answer 2

Suppose that you have experimental data `(x_i,y_i)` that you believe is related by an exponential function of the form:

`y = Ce^(kt)`

where `C` and `k` are constants, and `e` is the base of Natural logarithms (Euler's Number `2.71828182...`. Here if `k>` we would be modelling exponential growth (eg bacteria growth), and `k<0` would be exponential decay (eg radioactivity decay).

(NB Equality any equation of the form `y = CA^(kt) + D` can be written in the above form, so that exponential base we choose is arbitrary, and any additional constants can be incorporated into a single multiplier constant).

Then by taking Natural logarithms of both sides we get:

`lny = ln{Ce^(kt)}`
`" " = lnC + lne^(kt)`
`" " = lnC + kt`

If we write this as:

`lny = kt + lnC`

And compare with the standard equation of a straight lie@

`Y = mX +c`

Hopefully is clear that if we plot `lny` on the `y`-axis against `t` on the `x`-axis and draw a line of best fit, then `k=m` (the slope) and `lnC=c` (the `y`-intercept).

Obviously if plotting the data, does not produce a clear line of best fit then the original assumption about the data being modelled by an exponential function `y = Ce^(kt)` is incorrect.

On other note of caution, In mathematics we would generally choose base `e` because of the calculus, but often in physics base `10` is used because of scientific notation. As indicated above the base is arbitrary, choose the appropriate base and stick with it.

If you would like a solid example let me know and I will find a physics/maths exam question.