Section 4: Logarithmic models (the inverse model for exponential data)
The last basic models we will introduce in this course are logarithmic models. These are based on the LOG spreadsheet function discussed in an earlier topic. The main reason that logarithmic models are useful is that they are the inverse of exponential models—for a logarithmic model, equal-percentage changes in the input variable x result in equal-amount steps in the output variable y.
An example of a common use of a logarithmic model is in computing how old an archeological object is from how much radioactive Carbon-14 remains in it. The radioactivity of several objects of known age (such as tree rings) is measured, and then a model that predicts age from radioactivity level is fit to this data. Finally, measurements are made of the radioactivity of other similar objects (such as wooden tools) for which the age is unknown, and the model is used to estimate their age. (If you wanted to predict the level of radioactivity from the age you would use an exponential model, but here the roles of the variables are reversed.)
The two parameters of a logarithmic model are closely related to those for exponential models. The growth rate (or decay rate, if negative) parameter is the same as for exponential models, although in this case it is used as the second parameter (the logarithmic base) of the spreadsheet LOG function. Since the roles of x and y have been reversed, the initial-value (i.e., y-intercept) parameter from the exponential model become an x-intercept parameter in the logarithmic model. While there are several different ways that a logarithmic modeling formula could be written, the most convenient for our purposes will be , which implies a formula in C3 of “=LOG(A3/$G$3,1+$G$4)”.
Example 5: For a particular measurement apparatus and technique, radiation readings of 1274, 1085, 865, 697, and 276 were observed for objects with known ages of about 1100, 2400, 4300, 6100, and 13800 years, respectively. Use this data to create a model that predicts the age of similar objects, and use that model to estimate the age of such an object whose radiation measurement is 515.
[i] First, we must make an appropriate data table. Since we will be using the radiation readings to predict age, we will put the radiation readings into the column A and the respective age values to the right in column B.
[ii] Make a modeling spreadsheet, setting C3 to =LOG(A3/$G$3,1+$G$4) and spreading that model formula down to row 7 beside the data. Put the labels “x-intercept” and “rate” to the right of the parameter cells G3 and G4.
[iii] As usual, include deviations in column D and squared deviations in column E, and enter a sum-of-squared-deviations formula into a cell such as H8.
[iv] Adjust the parameters in G3 and G4 to initial values that make the model graph roughly match the data (example: G3 = 2000 & G4 = −0.0001)
[v] Use Solver to find the best-fit parameters: G3 = 1451 and G4 = −0.0001203; the implied model formula is age = LOG(radiation/1451, 1−0.0001203).
[vi] Enter the radiation level 515 into A8, giving an estimated age of 8,612 years.
NOTE: What is the real-world meaning of these model parameters? The growth/decay rate of −0.0001203 implies that 0.01203% of the Carbon-14 atoms decay each year. The “x-intercept” parameter shows what radiation-reading input would give an age output of zero; that is, what the radiation reading would be for a brand-new object.