# Ch 12: Exponential and Logarithmic Functions

### About This Chapter

## Exponential and Logarithmic Functions - Chapter Summary and Learning Objectives

Exponential and logarithmic functions deal with variables that are proportional to the functions' current values. The lessons in this chapter will help you understand how they work and about exponential growth and exponential decay. You'll learn methods of writing logarithmic equations and how to graph a logarithmic function. This chapter is designed to teach you to:

- Define an exponential function
- Define natural base e and graph its functions
- Evaluate a logarithm
- Graph a logarithmic function
- Solve exponential equations
- Calculate growth

Video | Objective |
---|---|

What is an Exponential Function? | Define an exponential function. |

Exponential Growth vs. Decay | Compare the processes of growth and decay. |

Transformation of Exponential Functions | Translate and exponential function horizontally and vertically. |

Using the Natural Base e | Define the nature base and graph natural base functions. |

What is a Logarithm? | Define a logarithm. |

How to Evaluate Logarithms | Evaluate logarithms. |

Inverse of Logarithmic Functions | Rewrite logarithmic equations in exponential form and vice versa. |

Exponentials, Logarithms and the Natural Logarithm | Understand and use the natural logarithm. |

Basic Graphs and Shifted Graphs of Logarithmic Functions | Graph a logarithmic function and identify its vertical asymptote, domain and range. |

Logarithmic Properties | Understand the basic properties of logarithms. |

Practice Problems for Logarithmic Properties | Learn to solve problems with logarithmic properties. |

How to Solve Logarithmic Equations | Learn to solve logarithmic equations. |

Change-of-Base Formula for Logarithms | Define the change-of-base formula and use it to evaluate logarithmic expressions. |

### 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.

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### Other Chapters

Other chapters within the Math 105: Precalculus Algebra course

- Mathematical Modeling
- Linear Equations & Inequalities
- Quadratic Functions
- Rational Expressions and Functions
- Polynomial Functions of a Higher Degree
- Absolute Value Equations & Inequalities
- Complex Numbers
- Geometry Basics
- Functions Overview
- Function Operations
- Graph Symmetry
- Introduction to the Derivative
- Studying for Math 105