# Ch 13: Exponentials & Logs

### About This Chapter

## Exponentials and Logs - Chapter Summary

In this chapter, you'll learn how to define and discuss the mathematical properties of exponentials and logarithms, graph exponential and logarithmic functions and solve equations related to money and interest. You'll also learn how to define negative and zero exponents, distinguish between linear and exponential growth patterns and describe the dramatic potential of exponential growth. In addition, the instructor will discuss real world exponential growth and the relationship between growth and decay.

During the course of the lesson, you can learn how to locate an exponential equation from a description, graph or point, as well as understand the coefficients of an equation in context. Additional topics include exponential and logarithmic functions and logarithmic scales. At the end of the chapter, you should be able to:

- Identify and work with representations of exponential and logarithmic functions, including algebraic, graphical, numerical and written forms
- Make and use the connections found among functions related to exponential or logarithmic expressions, as well as graphs and symbolic representations of functions
- Understand the relationship between exponential and logarithmic functions
- Use the laws and properties of exponents and logarithms to simplify expressions and solve problems
- Utilize mathematical representations, including graphs, numerical methods and tables, to solve equations, inequalities and systems that include exponential and logarithmic functions
- Model and solve problems involving exponential growth and decay
- Use logarithmic scales, such as decibel and Richter, to describe events and solve problems
- Model and solve finance-related math problems, like compound interest, using exponential and logarithmic functions.

Our concise and engaging video lessons on exponentials and logs are designed to expedite the learning process for busy educators and students. These easy-to-follow presentations are taught by experienced and knowledgeable instructors and are accompanied by lesson transcripts. Each tutorial also includes an online self-assessment quiz that can help viewers determine how well they understood the material. Through the use of video tags, major points can be revisited as often as needed, without having to re-watch the entire presentation.

### 1. What Are the Five Main Exponent Properties?

We'll look at the five important exponent properties and an example of each. You can think of them as the order of operations for exponents. Learn how to handle math problems with exponents here!

### 2. How to Define a Zero and Negative Exponent

The zero and negative exponent properties are two you will use quite a lot in mathematics. The negative exponent property can be confusing, but when you remember a couple fun ideas, you will get it right every time!

### 3. 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!

### 4. 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!

### 5. Writing & Graphing Exponential Functions

In this lesson, we will explore exponential functions and their properties. Through real world examples and explanation, we will see how to write and graph these types of functions.

### 6. Representations of Exponential & Logarithmic Functions

Quantitatively, a logarithmic function has as its inverse an exponential function. Qualitatively, we will show this inverse relationship both algebraically and graphically.

### 7. 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.

### 8. 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!

### 9. 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.

### 10. 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.

### 11. How to Graph Logarithms: Transformations and Effects on Domain/Range

The graph of a basic logarithm is relatively simple. This lesson will show you how to graph a logarithm and what the transformations will do to the graph as well as their effects on the domain and range of the graph.

### 12. Logarithmic Scale: Definition & Overview

A logarithmic scale is a useful way to represent information on a graph, especially when there are exponential changes in the magnitude of the numbers. Learn more, look at several examples and take a quiz to test your knowledge.

### 13. Using Exponential & Logarithmic Functions to Solve Finance Problems

Exponential and logarithmic functions can be seen in mathematical concepts in finance, specifically in compound interest. This relationship is illustrated by the exponential function and its natural logarithmic inverse.

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### Other Chapters

Other chapters within the TExES Mathematics 7-12 (235): Practice & Study Guide course

- About the TExES Math 7-12 Exam
- Real Numbers
- Mathematical Models
- Complex Numbers & the Complex Plane
- Number Theory
- Number Patterns
- Functions and Graphs
- Linear Functions
- Quadratic Functions & Polynomials
- Evaluating Piecewise & Composite Functions
- Rational and Radical Functions
- Inequalities and Absolute Values
- The Unit Circle
- Trigonometric Functions
- Using a Scientific Calculator for Calculus
- Understanding Limits in Math
- Understanding Rate of Change
- Calculating Derivatives of Functions
- Derivatives and Graphs
- Optimization in Calculus
- Definite Integrals and Sums
- Integration Applications in Calculus
- Working with Measurement
- Finding Volume, Area & Perimeter
- Introduction to Proofs and Constructions
- Congruence and Similarity
- Triangles
- Circles
- Real World Shapes
- Coordinate Geometry
- Understanding Transformations in Math
- Conic Sections
- Understanding Vectors
- Measuring & Displaying Data
- Data Distribution Overview
- Sampling in Statistics
- Distribution & Inference in Statistics
- Inference About a Mean
- Regression and Correlation
- Finding Probability
- Probability Distributions and Statistical Inference
- Experiments and Surveys
- Mathematical Process & Perspectives
- Teaching Strategies & Activities for the Math Classroom
- Differentiated Instructional Strategies for the Math Classroom
- Using Student Assessments in the Math Classroom
- TExES Mathematics 7-12 Flashcards