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

Mar 16, 2017

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:
$y = A {e}^{k x}$
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:
$5 = A {e}^{0 x}$ where ${e}^{0} = 1$
So we get $A = 5$

And

$100 = A {e}^{10 k}$ but $A = 5$ so:
$100 = 5 {e}^{10 k}$
Rearranging:
${e}^{10 k} = \frac{100}{5}$
${e}^{10 k} = 20$
Take the natural log of both sides to get rid of the exponential:
$10 k = \ln \left(20\right)$
$k = \ln \frac{20}{10} = 0.3$

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

$y = 5 {e}^{0.3 x}$

Mar 16, 2017

Suppose that you have experimental data $\left({x}_{i} , {y}_{i}\right)$ that you believe is related by an exponential function of the form:

$y = C {e}^{k t}$

where $C$ and $k$ are constants, and $e$ is the base of Natural logarithms (Euler's Number $2.71828182 \ldots$. 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 $y = C {A}^{k t} + D$ 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:

$\ln y = \ln \left\{C {e}^{k t}\right\}$
$\text{ } = \ln C + \ln {e}^{k t}$
$\text{ } = \ln C + k t$

If we write this as:

$\ln y = k t + \ln C$

And compare with the standard equation of a straight lie@

$Y = m X + c$

Hopefully is clear that if we plot $\ln y$ on the $y$-axis against $t$ on the $x$-axis and draw a line of best fit, then $k = m$ (the slope) and $\ln C = c$ (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 $y = C {e}^{k t}$ 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.