OVERVIEW: If a scatterplot shows a curved pattern, itcan perhaps be conveniently modeled by an exponential growthor decay function of the form
y = ab x or a power function of the form
y = ax b In these situations, we can linearize the data by making use oflogarithms. Among the advantages of using logarithms is the fact thatuse of logarithms produces smaller numbers, making graphical displaysmore convenient to construct.
Definition: log
Rules for logarithms
1. log(AB) = logA+ logB
2. log(A/B) = logA - logB
3. logAp = plogA
Note that y = abx (exponential function)
==> logy = loga + logbx
==> logy = loga + xlogb.
This is a linear relationship between the variables x andlogy since loga and log b are constants.
Also, y = axb (power function)
==> logy = loga + logxb
==> logy = loga + blogx.
This is a linear relationship between the variables logxand logy.
Here is a simple example using four points. (Use TI-83 to verifycalculations.)
x | y | logy | logx |
2 | 3 | 0.47712 | 0.30103 |
7 | 41 | 1.6128 | 0.8451 |
10 | 168 | 2.2253 | 1.0000 |
16 | 625 | 2.7959 | 1.2041 |
The points {(x,y)} form a curved pattern.
Fitting a least squares regression line to {(x,y)} yields
y (hat) = -186.6715 + 45.2482x
r2 = .8573, r = .9259
For x = 12, the predicted y value is y(hat) =-186.6715+45.2482(12) = 356.31.
Fitting an exponential function to {(x,y)} yields
y (hat) = (2.1347)(1.4640)x
r2 = .9526, r = .9760
For x = 12, the predicted value is y(hat) =(2.1347)(1.4640)12 = 206.93
Fitting a least squares regression line to {(x,logy)}, weget
log[y(hat)] = 0.3293 + 0.1655x
r2 = .9526, r = .9760
For x = 12, we have logy = 2.3153, and y(hat) =102.3153 = 206.68
Fitting a power function to {(x,y)} yields
y(hat) = (0.4416)x2.5464
r2 = .9843, r = .9921
For x = 12, y(hat) = (0.4416)(12)2.5464 = 247.20
Fitting a least squares regression line to {(logx,logy)}yields
log[y(hat)] = -0.3550 + (2.5464)logx
r2 = .9843, r = .9921
For x = 12, we have log[y(hat)] = -0.3550 + (2.5464)(log12) =2.3930, and y(hat) = 102.3930 = 247.19
Don't forget residuals. These are useful in determining thebest model to fit to a data set.
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