Modeling with Exponential & Logarithmic Functions

Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

This lesson will give a quick review of exponential and logarithmic functions and when to use these functions to model data. Then we will explore the use of graphing calculators in finding exponential and logarithmic functions to model given data.

Exponential and Logarithmic Functions

Suppose you just adopted two adorable bunnies from the local animal shelter. Over the next four months, you realize that they are reproducing so often that the number of bunnies you have seems to double each month.

Month # of Bunnies
1 2
2 4
3 8
4 16

How many bunnies will you have by the end of the year? To consider this, you plot the points you have in your table and see if you can come up with any estimates.


modexplog1


You see that the bunny population starts off rising quite slowly, but then more quickly. When data points take on this type of pattern, it indicates that the data should be modeled using an exponential function. Exponential functions are functions containing the variable in the exponent, and have a general form y = abx.

Hmm, how many months will have passed to get to 100 bunnies? You make another table with the number of months based on the number of bunnies.

# of Bunnies Month
2 1
4 2
8 3
16 4

Once again, you plot the points on a graph.


modexplog2


You see that the number of months increases quickly at first, but then more slowly. When data points take on a pattern of increasing quickly and then more slowly, a logarithmic function should be used to model the data.

A logarithmic function contains the variable within a logarithm, and has general form y = alogb (x).

Great! We see what types of models should be used to model the data in each instance, and when to use exponential or logarithmic models in general, but how in the world do we come up with these models? Graphing calculators to the rescue!

Modeling Using a Calculator

When we are given a set of data points that take on the pattern of an exponential or logarithmic function, we can use a graphing calculator to find the model that best fits the data, and use it to analyze the data.

Depending on your graphing calculator, the process may be a bit different. The following steps are based on using a Texas Instruments graphing calculator.

  1. To enter your data, Hit STAT, and choose Edit.
  2. Enter x-values in list L1, and y-values in list L2
  3. To find the model: Hit STAT, scroll to CALC, then choose ExpReg (for exponential models) or LnReg (for logarithmic models).
  4. The calculator needs to know where to find the data. Hit the '2nd' button, then 1, then comma (,). Hit the '2nd' button again, then 2. You'll see either ExpReg L1, L2 or LnReg L1,L2.
  5. Hit Enter for the model and values.

Okay, there are several buttons to push, but we can do this! Let's take our bunny data through these steps.

Exponential Model

Starting with the exponential model, we enter the data into our calculator's lists.

L1 L2
1 2
2 4
3 8
4 16

Next, we hit STAT, and go to our CALC menu. Since this is an exponential model, we scroll down to choose 'ExpReg'. We're brought back to the main screen, so we tell the calculator where to get the data from by entering L1 and L2, separated by a comma.

  • ExpReg L1, L2

Lastly, we hit enter. The screen gives us the exponential model and the values of a and b.

ExpReg

y = a*b^x
a = 1
b = 2

This tells us that the function we want to use to model the data is y = 1 ⋅ 22, or y = 2x, where y is the number of bunnies, and x is the number of months. Well, that's not so hard!

Logarithmic Model

Let's take a look at the logarithmic example. It's the exact same process, except that we enter different values in our lists.

L1 L2
2 1
4 2
8 3
16 4

and we chose 'LnReg' instead of 'ExpReg'. When we do this, the end result should display the following:

LnReg

y = a + blnx
a = 0
b = 1.442695041

Therefore, the function we should use to model the second set of data is y = 0 + 1.4427ln(x), or y = 1.4427ln(x), where y is the number of months, x is the number of bunnies, and ln is a logarithm with base e ≈ 2.71828

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