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

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Answer 1 · 2015-10-09T22:14:34

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

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.