4: Exponential and Logarithmic 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. 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.
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- 4.0: Introduction 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.
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- 4.1: Exponential Functions
- This section introduces exponential functions, focusing on their definition, properties, and applications. It explains how to identify exponential growth and decay, interpret graphs, and analyze their behavior. Examples demonstrate how to evaluate and use exponential functions in various contexts, such as modeling population growth or radioactive decay.
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- 4.2: Graphs of Exponential Functions
- This section explores the graphs of exponential functions, detailing key features such as domain, range, asymptotes, and intercepts. It explains how transformations like shifts, reflections, and stretches affect the shape of the graph. The section includes examples demonstrating how to graph exponential functions and analyze their behavior.
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- 4.3: Logarithmic Functions
- This section introduces logarithmic functions as the inverses of exponential functions. It covers their properties, common and natural logarithms, and how to evaluate and rewrite logarithmic expressions. The section also explains the relationship between logarithmic and exponential equations, including conversion between forms. Examples illustrate solving logarithmic equations and their real-world applications.
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- 4.4: Graphs of Logarithmic Functions
- This section explores the graphs of logarithmic functions, explaining their key features such as domain, range, vertical asymptotes, and intercepts. It discusses how transformations, including shifts, reflections, and stretches, affect the graph. Examples illustrate how to sketch logarithmic functions and analyze their behavior in different contexts.
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- 4.5: Logarithmic Properties
- This section explores logarithmic properties, including the product, quotient, and power rules, which simplify logarithmic expressions. It also introduces the change-of-base formula, allowing logarithms to be rewritten in different bases. Examples illustrate how to apply these properties to simplify expressions and solve logarithmic equations.
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- 4.6: Exponential and Logarithmic Equations
- This section covers solving exponential and logarithmic equations using algebraic techniques, properties of exponents and logarithms, and logarithmic conversions. It explains how to apply logarithms to isolate variables, use the one-to-one property, and handle real-world applications like exponential growth and decay. Examples illustrate step-by-step solutions for different types of equations.
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- 4.7: Exponential and Logarithmic Models
- This section explores real-world applications of exponential and logarithmic functions, including population growth, radioactive decay, carbon-14 dating, logistic growth, and Newton’s Law of Cooling. It explains key concepts such as doubling time and half-life, showing how these models are used in scientific and financial contexts. Examples illustrate how to apply these functions to solve practical problems.
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).