Back To Course

Math 101: College Algebra12 chapters | 95 lessons | 11 flashcard sets

Are you a student or a teacher?

Start Your Free Trial To Continue Watching

As a member, you'll also get unlimited access to over 70,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.

Free 5-day trial
Your next lesson will play in
10 seconds

Lesson Transcript

Instructor:
*Yuanxin (Amy) Yang Alcocer*

Amy has a master's degree in secondary education and has taught math at a public charter high school.

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.

Solving for your variable, usually *x*, is pretty straightforward when your *x* can easily be isolated through addition, subtraction, multiplication, or division. But if your variable is an exponent, then it becomes a bit harder to solve for it. These types of problems are called **exponential equations**. Here are some examples of exponential equations:

*e**x* = 40

10*x* + 2 = 30

*e**2x* - 7*e**x* + 10 = 0

At first look, it may seem that isolating the *x* is a difficult, if not impossible, task to do. That is, until you realize that you can use the logarithmic function to help you isolate the *x*. Let's see how you can do that.

The **log function** is the inverse of an exponential function. It has a very useful property that can help you solve for your exponential equations.

*x* = *b**y* with the function *y* = log*b**x*

This means that if the variable you are trying to solve for is in the exponent, then you can use the logarithm function, like this:

(log*b*)(*x**y*) = *y*log*b**x*

Let's see if you can use this to help you solve this exponential equation:

10*x* + 2 = 30

Even though the *x*, the variable you are trying to solve for, is in the exponent, you are still going to use **algebra techniques** to solve this problem. This means trying to isolate your variable to the best of your ability. So you first subtract the 2 from both sides. Remember, whatever you do on one side, you have to do to the other as well.

10*x* + 2 - 2 = 30 - 2

10*x* = 28

Now you can take the log of both sides to help you move your variable out of the exponent so you can solve for it.

log(10*x*) = 28

You can now use the useful property of the logarithm function to move your *x* out of the exponent area.

(log*b*)(*x**y*) = *y*log*b**x*

Applying this property, you get this:

*x* log (10) = log (28)

*x* = log (28)

*x* = 1.447

And you are done! You have solved the problem!

For some problems, you might see a little *e*. This *e* is called **Euler's number** and is approximately equal to 2.71828. If you see an exponential equation with this number in it, then instead of using the log function, you'll use the **natural log function ln**, since the natural log has a base of *e*. You can use the natural log when your bases match. So if you have an *e*, then that means you have a base of *e*. If you see a 10 or any other number, then your base is a 10 and the standard log function will work.

The process of solving an exponential equation with an *e* in it instead of a base 10 number is the same process, except you're now using the natural log function. Let's try solving this problem:

*e**x* = 40

First you'll take the natural log of both sides.

ln *e**x* = ln (40)

Now you can move your *x* out of the exponent area to get this:

*x* ln *e* = ln (40)

*x* = ln (40)

*x* = 3.689

And you are done! Let's try solving another problem.

*e**2x* - 7*e**x* + 10 = 0

This one is a bit more interesting. Instead of having just one term with the *x* in the variable, you now have two of them. To solve this, you'll need to go about it in a different way. Looking closer at your numbers, you see that you can actually go ahead and factor your problem to help you solve it, like this:

*e**2x* - 7*e**x* +10 = 0

(*e**x* - 2)(*e**x* - 5) = 0

Once you've factored, now you can go ahead and solve each of your factors using the log method like you've done before.

*e**x* - 2 = 0

*e**x* = 2

ln *e**x* = ln (2)

*x* = ln (2)

*x* = 0.69

That's one solution. The other one is this:

*e**x* - 5 = 0

*e**x* = 5

ln *e**x* = ln (5)

*x* = ln (5)

*x* = 1.609

And you are done!

Let's review. An **exponential equation** is a problem with exponents that usually has a variable in the exponent area that you have to solve for. To solve these types of problems, you'll make use of the **logarithmic function**.

*x* = *b**y* with the function is *y* = log*b**x*, along with this property: log*b**x**y* = *y*log*b**x*

This property only applies when your bases are the same. Use the standard log with a base of 10 when working with base 10 numbers. If you see an *e*, then use the natural logarithm, the logarithm with a base of *e*. You'll use the same methods that you use to solve all your other algebra problems with the addition of the log function helping you to take the variable out of the exponent.

To unlock this lesson you must be a Study.com Member.

Create your account

Are you a student or a teacher?

Already a member? Log In

BackDid you know… We have over 160 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

You are viewing lesson
Lesson
7 in chapter 9 of the course:

Back To Course

Math 101: College Algebra12 chapters | 95 lessons | 11 flashcard sets

- Chemistry 304: Organic Chemistry II
- ILTS Information Guide
- Introduction to Cloud Computing
- Praxis Elementary Education - Content Knowledge (5018): Study Guide & Test Prep
- CTEL Information Guide
- Atomic Structure in Chemistry Basics
- Equilibrium in Chemistry Basics
- Weathering, Erosion & Wasting Basics
- Overview of Waves, Sound & Light
- Overview of Types of Reactions in Chemistry
- Texas Teacher Certification Test Limit Waiver
- AFOQT Cost
- What Does the HESI A2 Nursing Exam Consist of?
- How to Learn Pharmacology for NCLEX
- What Are Considered Higher-Level Questions on the NCLEX?
- How to Study for NCLEx in 2 Weeks
- How Hard Is the ASVAB

- The Teacher's Role in Socializing Students to Be Physically Active
- Addressing Different Learning Styles in PE Programs
- Cardiorespiratory Fitness Activities for Various Levels
- First Language vs. Second Language Acquisition
- Regular Expressions in SQL Server Databases: Implementation & Use
- IPv6 Address: Impact, Structure & Types
- Real-World Implementations of Cloud Computing
- Network Switching: Definition & Types
- Quiz & Worksheet - Space Awareness & Body Movement
- Quiz & Worksheet - Boy by Roald Dahl
- Quiz & Worksheet - Sexual Frustration
- Quiz & Worksheet - Teenage Heart Attack Symptoms
- Quiz & Worksheet - Aerobic Endurance
- Flashcards - Measurement & Experimental Design
- Flashcards - Stars & Celestial Bodies

- Business Law Textbook
- Common Core Math Grade 8 - The Number System: Standards
- Building & Launching a Successful Team
- Holt World History - Human Legacy: Online Textbook Help
- Common Core ELA - Informational Text Grades 11-12: Standards
- NMTA Social Science: Ethnicity & Geography
- GACE Middle Grades ELA: The Writing Process
- Quiz & Worksheet - The Multiplier Effect and Simple Spending Multiplier
- Quiz & Worksheet - Characteristics of the Human Relations Movement
- Quiz & Worksheet - How to Write the Inverse of Logarithmic Functions
- Quiz & Worksheet - Detail & Word Meaning in ACT Reading
- Quiz & Worksheet - Emotional Labor & Dissonance in the Workplace

- The Theory of Attribution In Organizational Behavior: Definition and Three Determinants
- The Westing Game: Summary & Characters
- How to Teach English Abroad
- Narrative Writing Rubric Examples
- How to Pass an Excel Test
- Curriculum Development Templates
- Day of the Dead Lesson Plan
- ACT Accommodations for ELL Students
- Actuarial Science Exams
- Poetry Books & Activities for Kindergarten
- Adult Community Education
- 9th Grade Reading List

- Tech and Engineering - Videos
- Tech and Engineering - Quizzes
- Tech and Engineering - Questions & Answers

Browse by subject