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6: Exponential and Logarithmic Functions

Last updated

Dec 26, 2024
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- 5.8E: Modeling Using Variation (Exercises)
- 6.0: Prelude to Exponential and Logarithmic Functions

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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.

6.0: 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.
6.1: 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.
- 6.1E: Exponential Functions (Exercises)
6.2: Graphs of Exponential Functions
As we discussed in the previous section, exponential functions are used for many real-world applications such as finance, forensics, computer science, and most of the life sciences. 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.
- 6.2E: Graphs of Exponential Functions (Exercises)
6.3: Logarithmic Functions
The inverse of an exponential function is a logarithmic function, and the inverse of a logarithmic function is an exponential function.
- 6.3E: Logarithmic Functions (Exercises)
6.4: 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.
- 6.4E: Graphs of Logarithmic Functions (Exercises)
6.5: 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.
- 6.5E: Logarithmic Properties (Exercises)
6.6: 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.
- 6.6E: Exponential and Logarithmic Equations (Exercises)
6.7: 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.
- 6.7E: Exponential and Logarithmic Models (Exercises)
6.8: Fitting Exponential Models to Data
We will concentrate on three types of regression models in this section: exponential, logarithmic, and logistic. Having already worked with each of these functions gives us an advantage. Knowing their formal definitions, the behavior of their graphs, and some of their real-world applications gives us the opportunity to deepen our understanding. As each regression model is presented, key features and definitions of its associated function are included for review.
- 6.8E: Fitting Exponential Models to Data (Exercises)

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

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