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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.

- 4.0: Prelude to Exponential and Logarithmic Functions
- 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.

- 4.1: Composition of Functions
- Suppose we want to calculate how much it costs to heat a house on a particular day of the year. The cost to heat a house will depend on the average daily temperature, and in turn, the average daily temperature depends on the particular day of the year. The cost depends on the temperature, and the temperature depends on the day. By combining these two relationships into one function, we have performed function composition, which is the focus of this section.

- 4.2: Inverse Functions
- If some physical machines can run in two directions, we might ask whether some of the function “machines” we have been studying can also run backwards. In this section, we will consider the reverse nature of functions.

- 4.3: Exponential Functions
- When populations grow rapidly, we often say that the growth is “exponential,” meaning that something is growing very rapidly. To a mathematician, however, the term exponential growth has a very specific meaning. In this section, we will take a look at exponential functions, which model this kind of rapid growth.

- 4.4: Graphs of Exponential Functions
- Working with an equation that describes a real-world situation gives us a method for making predictions. Most of the time, however, the equation itself is not enough. We learn a lot about things by seeing their pictorial representations, and that is exactly why graphing exponential equations is a powerful tool. It gives us another layer of insight for predicting future events.

- 4.5: Logarithmic Functions
- The inverse of an exponential function is a logarithmic function, and the inverse of a logarithmic function is an exponential function.

- 4.6: Graphs of Logarithmic Functions
- In this section we will discuss the values for which a logarithmic function is defined, and then turn our attention to graphing the family of logarithmic functions.

- 4.7: Logarithmic Properties
- Recall that the logarithmic and exponential functions “undo” each other. This means that logarithms have similar properties to exponents. Some important properties of logarithms are given here.

- 4.8: Exponential and Logarithmic Equations
- Uncontrolled population growth can be modeled with exponential functions. Equations resulting from those exponential functions can be solved to analyze and make predictions about exponential growth. In this section, we will learn techniques for solving exponential functions.

- 4.9: Exponential and Logarithmic Models
- We have already explored some basic applications of exponential and logarithmic functions. In this section, we explore some important applications in more depth, including radioactive isotopes and Newton’s Law of Cooling.

Thumbnail: The functions \(y=e^x\) and \(y=\ln(x)\) are inverses of each other, so their graphs are symmetric about the line \(y=x\).

## Contributors

Jay Abramson (Arizona State University) with contributing authors. Textbook content produced byOpenStax Collegeis licensed under aCreative Commons Attribution License 4.0license.Download for free athttps://openstax.org/details/books/precalculus.