Developing Equations from Empirical Data Which Fits the Exponential Function

Exponential Form

y=bemx

Review Logarithms and Exponents

see Link or Text Page 1046

Method 1
Method 2

y=bemx

Take the natural logarithm of both sides of the exponential function and you get

ln y = ln bemx

ln y = ln b + ln emx

ln y = ln b + mx

This is in the form

y = mx + b

Now you can see that if you plot the ln y vs the x you will get a straight line with a slope of m.

y=bemx

Alternatively

y/b=emx

ln (y/b) = mx

This form is the most convenient when it is previously known that the data fits an exponential function. (a straight line graph using method 1 or 2 below usually indicates exponential growth or decay.)

for example

N=Noekt

typically No would be an initial number (of things or observations at time, t=0) and N would be the number at some future time, t.

then

ln (N/No) = kt

to resolve k we only need the ratio of N/No

Example Data: p - population of insects in a field, t - time

t
p
0
1.26 x 106
1.00
6.38 x 105
2 .00
3.23 x 105
3 .00
1.63 x 105
5 .00
4.21 x 104
7 .00
1.08 x 104
1.00 x10
1.40 x 103

A normal graph does not easily give the constants in the power function

Solution

  • In both approaches a graph is drawn to verify that the data fits an equation in the form, y=bemx. If a straight line is formed this verifies that the data should fit an equation in the form y=bemx
  • In both approaches, two points are selected and applied to the equation in the form ln y = ln b + mx, (which forms two equations two unknowns). Solve for m and b, then apply back to the equation in the original form,y=bemx . This is the answer.

 

Approach #1

Find ln p and plot on regular graph paper against t.(link to internet source of graph paper) Select two points on the line and substitute into the equation in the form, ln y = ln b + mx, and solve for m and b to get the equation.

Example Solution to approach #1

Approach #2

Plot x and y directly on semi log paper. (link to internet source of graph paper) Select two points on the line and substitute into the equation in the form, ln y = ln b + mx, and solve for m and b to get the equation. Take careful note that numbers are the variables and not the ln of the variables.

Example Solution to approach #2
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