Developing Equations from Empirical Data Which Fits the Power Function

Power Function Form
y=bxm

Review Logarithms and Exponents

see Link or Text Page 1046

Take the log of both sides of the power function and you get

log y = log b + m log x

This is in the form

y = mx + b

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

Example Data

x
y
1.00
4.18
2.00
3.35 x 10
3.00
1.13 x 102
4.00
2.68 x 102
5.00
5.23 x 102
8.00
2.14 x 103
1.00 x10
4.19 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=bxm. If a straight line is formed this verifies that the data should fit an equation in the form y=bxm
  • In both approaches, two points are selected and applied to the equation in the form log y = log b + m log x, (which forms two equations two unknowns). Solve for m and b, then apply back to the equation in the original form, y=bxm . This is the answer.

Approach #1 -

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

Example Solution to approach #1

Approach #2

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

Example Solution to approach #2
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