Formula
The exponential equation is `y=ax^b`
taking natural logarithm on both sides, we get
`ln(y)=ln(ax^b)`
`ln(y)=ln(a)+ln(x^b)`
`ln(y)=ln(a)+b ln(x)`
`Y=A+bX` where `Y=ln(y), A=ln(a), X=ln(x)`
which linear in Y,X
So the corresponding normal equations are
`sum Y = nA + b sum X`
`sum XY = A sum X + b sum X^2`
|
Examples
1. Calculate Fitting exponential equation `(y=ax^b)` - Curve fitting using Least square method
Solution:
The curve to be fitted is `y=ax^b`
taking logarithm on both sides, we get
`ln(y)=ln(a)+b ln(x)`
`Y=A+bX` where `Y=ln(y), A=ln(a), X=ln(x)`
which linear in Y,X
So the corresponding normal equations are
`sum Y = nA + b sum X`
`sum XY = A sum X + b sum X^2`
The values are calculated using the following table
| `x` | `y` | `X=ln(x)` | `Y=ln(y)` | `X^2` | `X*Y` |
| 2 | 27.8 | 0.6931 | 3.325 | 0.4805 | 2.3047 |
| 3 | 62.1 | 1.0986 | 4.1287 | 1.2069 | 4.5359 |
| 4 | 110 | 1.3863 | 4.7005 | 1.9218 | 6.5162 |
| 5 | 161 | 1.6094 | 5.0814 | 2.5903 | 8.1782 |
| --- | --- | --- | --- | --- | --- |
| `sum x=14` | `sum y=360.9` | `sum X=4.7875` | `sum Y=17.2357` | `sum X^2=6.1995` | `sum X*Y=21.5351` |
Substituting these values in the normal equations
`4A+4.7875b=17.2357`
`4.7875A+6.1995b=21.5351`
Solving these two equations using Elimination method,
`4a+4.7875b=17.2357`
and `4.7875a+6.1995b=21.5351`
`:.4.79a+6.2b=21.54`
`4a+4.7875b=17.2357 ->(1)`
`4.7875a+6.1995b=21.5351 ->(2)`
equation`(1) xx 4.7875 =>19.15a+22.920156b=82.515914`
equation`(2) xx 4 =>19.15a+24.798b=86.1404`
Substracting `=>-1.877844b=-3.624486`
`=>1.877844b=3.624486`
`=>b=3.624486/1.877844`
`=>b=1.930132`
Putting `b=1.930132` in equation `(1)`, we have
`4a+4.7875(1.930132)=17.2357`
`=>4a=17.2357-9.240507`
`=>4a=7.995193`
`=>a=7.995193/4`
`=>a=1.998798`
`:.a=1.998798" and "b=1.930132`
we obtain `A=1.9988,b=1.9301`
`:. a=antiln(A)=antiln(1.9988)=7.3802`
Now substituting this values in the equation is `y = a x^b`, we get
`y=7.3802*x^(1.9301)`
This material is intended as a summary. Use your textbook for detail explanation.
Any bug, improvement, feedback then
