About This Chapter
Exponential Functions and Logarithmic Functions - Chapter Summary and Learning Objectives
Did you know that exponential functions and logarithmic functions are inverses of each other? This means that if you can tackle one type of function, you'll likely be able to work with the other. Instructors teaching this chapter go over the ins and outs of both functions by covering such topics as logarithmic properties, Euler's number and the change-of-base formula. They also show you how to graph and solve exponential and logarithmic equations, and explore their uses to calculate exponential growth and decay as well as interest rates. Use this chapter to learn how to:
- Identify graphs of exponential functions
- Write and evaluate logarithmic functions
- Describe logarithmic properties
- Apply the change-of-base formula
|What Is an Exponential Function?||Review the types of graphs. Learn what the equation of an exponential function looks like.|
|Exponential Growth vs. Decay||Examine the graphs of exponential growth and exponential decay. Learn how to differentiate between the two.|
|Transformation of Exponential Functions: Examples, Lesson & Quiz||Follow the steps required to translate an exponential function vertically and horizontally. Learn how to graph reversals.|
|Using the Natural Base e: Definition, Lesson & Quiz||Explore characteristics of Euler's number. Find out what the graph of a natural log looks like, and examine its uses.|
|What Is a Logarithm?||Discover the relationship between logarithms and exponential functions.|
|How to Evaluate Logarithms||Review logarithmic form. Explore the uses of the change-of-base formula to evaluate logarithms without whole numbers in the solution.|
|Writing the Inverse of Logarithmic Functions: Lesson & Quiz||Find out how to change logarithmic equations into exponential form and back again.|
|Exponentials, Logarithms & the Natural Log||Examine the rules of the natural log alongside properties of exponentials. Explore some of their real-life applications.|
|Basic Graphs & Shifted Graphs of Logarithmic Functions: Definition & Examples||Understand how to graph a basic logarithmic function and perform transformations that flip or shift graphs vertically and horizontally.|
|Logarithmic Properties||Investigate the product, quotient and power properties of logarithms.|
|Practice Problems for Logarithmic Properties||Practice using logarithmic properties to expand, condense and evaluate logs.|
|How to Solve Logarithmic Equations||Survey the uses of inverse operations, the product property and exponentiating to solve logarithmic equations.|
|Using the Change-of-Base Formula for Logarithms: Definition & Example||Learn how the change-of-base formula can be used to evaluate logarithmic expressions with a base other than ten.|
|How to Solve Exponential Equations||Practice solving exponential equations by using logarithms and the change-of-base formula.|
|Calculating Rate and Exponential Growth: The Population Dynamics Problem||Learn how to write and solve differential equations useful for determining exponential growth.|
|Comparing the Large-Scale Behavior of Exponential and Logarithmic Functions with Different Bases||Examine how growth rates are represented on graphs of exponential and logarithmic functions.|
1. What Is an Exponential Function?
Brand new technologies don't always catch on right away because they can be expensive and don't always work as well as they should. But once the price comes down, and they start to work better, it doesn't take very long before it seems like everyone has one. Learn about the numbers behind this, exponential functions!
2. Exponential Growth vs. Decay
How is it that it only takes four years for our computer to go from top-of-the-line to almost worthless? Well, it has something to do with what's called exponential decay, which we'll learn about here!
3. Transformation of Exponential Functions: Examples & Summary
All exponential functions follow a basic graph. But when you make changes to the function, you will see the graph shift and make changes. Watch this video lesson to learn how to easily identify these changes or transformations.
4. Using the Natural Base e: Definition & Overview
In this video lesson, you will learn what kind of number 'e' is. Also learn the formal name for this unique number. See what kind of graph the natural base 'e' graphs into, as well as the uses of these types of functions.
5. What is a Logarithm?
Logarithms can help you solve exponential functions. In this lesson, you will learn about how to work with and recognize logarithms, and how to use logarithm notation with an example problem about earthquakes!
6. How to Evaluate Logarithms
Using this lesson, you can get practice evaluating logarithms, as well as learn some of the shortcuts behind writing and estimating them. You can also learn how to use your calculator to evaluate logarithms, and learn about a concept called the change of base theory.
7. Writing the Inverse of Logarithmic Functions
Watch this video lesson to learn how inverses are related to the original function. Also learn a method to find the inverse of logarithmic functions that you can easily use.
8. Exponentials, Logarithms & the Natural Log
Use the properties of exponentials and logarithms to learn how carbon dating works. This lesson covers properties of a natural log and rules of logarithms.
9. Basic Graphs & Shifted Graphs of Logarithmic Functions: Definition & Examples
Watch this video lesson and you will see what the basic graph of the logarithmic function looks like. You will also be able to identify the kind of shifts that will occur by just looking at a logarithmic function when you finish the video.
10. Logarithmic Properties
Along with the change of base property, there are three other logarithmic properties that allow you to manipulate expressions to your advantage. Learn about the product rule, quotient rule and power rule here.
11. Practice Problems for Logarithmic Properties
As you get further and further into mathematics, logarithms will appear more and more. It's the point of this lesson to get you used to dealing with them and able to know what to do when you see a whole line of paper full of them!
12. How to Solve Logarithmic Equations
If you're ever picking out a telescope to see your favorite planet, make sure you do the math first! Learn how to solve the logarithmic equation that will get you the size telescope you want.
13. Using the Change-of-Base Formula for Logarithms: Definition & Example
Logarithms have many uses in the real world. But what happens when the calculation uses a base other than the standard base 10? Watch this video to find out how to handle this situation and the formula to use.
14. How to Solve Exponential Equations
Read this lesson to learn the steps you need to take to solve exponential equations. You'll see that it's not too difficult; you just need to make use of one other mathematical operation in order to solve your problems.
15. Calculating Rate and Exponential Growth: The Population Dynamics Problem
You know how the world population keeps increasing? It's increasing faster now than it was 100 or 1,000 years ago. In this lesson, learn how differential equations predict this type of exponential growth.
16. Behavior of Exponential and Logarithmic Functions
After watching this video lesson, you will be able to recognize exponential and logarithmic functions by looking at the end behavior of the graphs. You will also learn how the graphs change.
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Other chapters within the Precalculus: High School course
- Working With Inequalities
- Absolute Value Equations
- Working with Complex Numbers
- Introduction to Quadratics
- Working with Quadratic Functions
- Basics of Polynomial Functions
- Working with Higher-Degree Polynomials
- Graphing Piecewise Functions
- Understanding Function Operations
- Graph Symmetry
- Graphing with Functions
- Rate of Change
- Rational Functions & Difference Quotients
- Rational Expressions and Function Graphs
- Using Trigonometric Functions
- Trigonometric Graphs
- Trigonometric Applications
- Solving Trigonometric Identities
- Vectors, Matrices and Determinants
- Mathematical Sequences and Series
- Sets in Algebra
- Analytic Geometry and Conic Sections
- Polar Coordinates and Parameterizations
- Teacher Resources for Precalculus in High School