Why learn about exponential and logarithmic functions?

Joan decided to try online dating since she has a strict rule about not dating people at work, and the guys in her math class are all taken.  Not long after posting a profile on a site her friend recommended to her, she was asked out to dinner by a cute, young man who shared some of the same interests and music tastes as her.

Joan nervously prepared for her first date from an online dating site, they had planned to meet at a sushi restaurant. After getting lectured by her dad about how to be safe on a “blind” date, Joan went to meet what she hoped was her next dreamy boyfriend.

Things didn’t get off to a good start, as it seemed that he had lied about his height.  Joan is almost 6 feet tall and prefers to date men that are at least as tall as her. Maybe he can’t remember, she thought, wanting it to be OK.

The night got progressively worse for Joan from there.  Her date ordered drink after drink and his volume increased with every one.  The people around them began to get up and leave.  As they were getting ready to pay the bill he claimed to have forgotten his card. What a disaster!

Electron micrograph of E.Coli bacteria (credit: “Mattosaurus,” Wikimedia Commons)

Later that night, to add insult to injury, Joan found herself very ill with food poisoning from the sushi restaurant.

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, as in the case of Joan’s food poisoning. 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.^[1]

For simplicity’s sake, imagine that the bacterial cells that caused Joan’s food poisoning can divide every hour. The table below 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! No wonder it didn’t take long for Joan to feel sick after eating sushi with her awful date.

Exponential Functions

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 and viruses. 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.