Figure 1 electron micrograph of E. Coli bacteria (credit: “mattosaurus,” Wikimedia Commons)
6.1 exponential Functions 6.5 logarithmic Properties 6.2 graphs of exponential Functions 6.6 exponential and logarithmic equations 6.3 logarithmic Functions 6.7 exponential and logarithmic models 6.4 graphs of logarithmic Functions 6.8 Fitting exponential models to data
Introduction Focus in on a square centimeter of your skin. Look closer. Closer still. If you could look closely enough, you would see hundreds of thousands of microscopic organisms. They are bacteria, and they are not only on your skin, but in your mouth, nose, and even your intestines. In fact, the bacterial cells in your body at any given moment outnumber your own cells. But that is no reason to feel bad about yourself. While some bacteria can cause illness, many are healthy and even essential to the body.
Bacteria commonly reproduce through a process called binary fission, during which one bacterial cell splits into two. When conditions are right, bacteria can reproduce very quickly. Unlike humans and other complex organisms, the time required to form a new generation of bacteria is often a matter of minutes or hours, as opposed to days or years.
For simplicity’s sake, suppose we begin with a culture of one bacterial cell that can divide every hour. Table 1 shows the number of bacterial cells at the end of each subsequent hour. We see that the single bacterial cell leads to over one thousand bacterial cells in just ten hours! And if we were to extrapolate the table to twenty-four hours, we would have over 16 million!
Hour 0 1 2 3 4 5 6 7 8 9 10 Bacteria 1 2 4 8 16 32 64 128 256 512 1024
In this chapter, we will explore exponential functions, which can be used for, among other things, modeling growth patterns such as those found in bacteria. We will also investigate logarithmic functions, which are closely related to exponential functions. Both types of functions have numerous real-world applications when it comes to modeling and interpreting data. 16 Todar, PhD, Kenneth. Todar’s Online Textbook of Bacteriology. http://textbookofbacteriology.net/growth_3.html.
Exponential and Logarithmic Functions
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476 CHAPTER 6 exPoNeNtial aNd logarithmic fuNctioNs
6.1 SeCTIOn exeRCISeS
veRbAl 1. Explain why the values of an increasing exponential
function will eventually overtake the values of an increasing linear function.
2. Given a formula for an exponential function, is it possible to determine whether the function grows or decays exponentially just by looking at the formula? Explain.
3. The Oxford Dictionary defines the word nominal as a value that is “stated or expressed but not necessarily corresponding exactly to the real value.” Develop a reasonable argument for why the term nominal rate is used to describe the annual percentage rate of an investment account that compounds interest.
AlgebRAIC For the following exercises, identify whether the statement represents an exponential function. Explain.
4. The average annual population increase of a pack of wolves is 25.
5. A population of bacteria decreases by a factor of 1 __ 8 every 24 hours.
6. The value of a coin collection has increased by 3.25% annually over the last 20 years.
7. For each training session, a personal trainer charges his clients $5 less than the previous training session.
8. The height of a projectile at time t is represented by the function h(t) = −4.9t 2 + 18t + 40.
For the following exercises, consider this scenario: For each year t, the population of a forest of trees is represented by the function A(t) = 115(1.025)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 82(1.029)t. (Round answers to the nearest whole number.)
9. Which forest’s population is growing at a faster rate? 10. Which forest had a greater number of trees initially? By how many?
11. Assuming the population growth models continue to represent the growth of the forests, which forest will have a greater number of trees after 20 years? By how many?
12. Assuming the population growth models continue to represent the growth of the forests, which forest will have a greater number of trees after 100 years? By how many?
13. Discuss the above results from the previous four exercises. Assuming the population growth models continue to represent the growth of the forests, which forest will have the greater number of trees in the long run? Why? What are some factors that might influence the long-term validity of the exponential growth model?
For the following exercises, determine whether the equation represents exponential growth, exponential decay, or neither. Explain.
14. y = 300(1 − t)5 15. y = 220(1.06)x
16. y = 16.5(1.025) 1 _ x 17. y = 11,701(0.97)t
For the following exercises, find the formula for an exponential function that passes through the two points given.
18. (0, 6) and (3, 750) 19. (0, 2000) and (2, 20) 20. −1, 3 _ 2 and (3, 24) 21. (−2, 6) and (3, 1) 22. (3, 1) and (5, 4)
18. Oxford Dictionary. http://oxforddictionaries.com/us/definition/american_english/nomina.