Question

How do you Find the exponential growth rate for a given data set?

Answer 1

The quick answer is: Take logs and find the slope.

Explanation:

If your data is exponential, then the log of the data will be linear.

If the data set is exact, then the log of the data set will exactly fit a line with equation `y = mx + c` or similar.

Slope `m` is given by the formula:

`m = (Delta y)/(Delta x) = (y_2 - y_1)/(x_2 - x_1)`

where `(x_1, y_1)` and `(x_2, y_2)` are two points on the line.

For example, consider the data:

`f(0) = 3`
`f(2) = 12`
`f(3) = 24`
`f(7) = 384`

Taking natural logs we find:

`ln(f(0)) ~~ 1.0968`
`ln(f(2)) ~~ 2.4849`
`ln(f(3)) ~~ 3.17805`
`ln(f(7)) ~~ 5.95064`

Hence slope `~~ 0.693` and `f(x) ~~ 3 * e^(0.693 x)`

`0.693` being the approximation of `ln(2)` that we found from the data. Actually `f(x) = 3*2^x = 3*e^(ln(2)x)`

If the data set is inexact (e.g. measurements from an experiment), then take logs before performing a linear regression or similar.