Back To Course

Math 101: College Algebra12 chapters | 94 lessons | 11 flashcard sets

Watch short & fun videos
**
Start Your Free Trial Today
**

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:
*Megan Banner*

If you're planning on putting some money in the bank to save up for that next thing you've got to have, it might be best to know a little about exponential equations first. Learn how to check how long it will take you to have as much money as you need here!

One of the most likely places for you to find an exponential equation in real life (and also on a standardized test) is in a bank when you're talking about saving money. When you are letting your money gain interest, chances are it's gaining interest exponentially. For example, every year you keep it in the bank, it grows 5% more. One equation that is often used to calculate how fast your money will grow is this: *A=Pert*

It can help us say things like, if you put $500 in a bank account that pays 6% interest, after 10 years, your money will have turned into $911. But if instead I wanted to know how long it would take for that $500 to become enough for the $4,000 motorcycle I've had my eye on, I'd be stuck with an exponential equation that needed solving.

Substituting the variables in the correct places in the equation would this time give me 4,000=500*e*0.06(*t*). The goal is to figure out what is *t*, or how many years it is going to take for the initial $500 to turn into the $4,000 I want it to be. I simply have to use inverse operations to undo things one at a time to try to get the *t* by itself. I undo multiplication with division, and I end up with this equation: 8=*e*0.06(*t*). But now we have to undo an exponential. The variable I'm trying to solve is stuck in the exponent.

Good thing we know about logs. In order to undo the base of an *e*, I have to take the log with the base of an *e*, and the log with the base of an *e* is called the natural log. So by taking the natural log of both sides, the natural log of *e* cancels out, or undoes, and I just get the 0.06(*t*). On the other side I have the natural log of 8.

Because the ln button is on my calculator, I don't have to use the change of base formula to evaluate it, and we can just plug it in to find that it equals about 2.079. Now I simply have to undo one more multiplication step with division, and I find that *t* would have to be about 34.66 years. I've got a ways to go.

As a quick note, when we use logs to undo exponentials that aren't the common ones on our calculator, say, undoing 8*x* = 50 with the log base 8 on both sides, we need to use the change of base formula to estimate our answer. So the log base 8 on the left cancels with the 8*x*, and we just get *x*, but on the right I have the log base 8(50). Because the log base 8 isn't a very common logarithm, it's not on our calculator, so we have to use the change of base formula. The change of base formula says I can turn the log base 8 into log base 10s being divided by each other. So I get the log base 10 (50) divided by the log base 10 (8), which I can plug into my calculator to find is about 1.88.

We can also solve exponential equations that have exponentials on both sides, like this: 5(2*x* - 1) = 5(3*x* + 5). Because I have variables stuck in exponents on either side, I would like to undo the exponential on both sides. Luckily, the base on both sides is 5, so they can be undone the same way. By taking the log base 5 of both sides, they both cancel out, and I simply get 2*x* - 1 = 3*x* + 5. Now this is just a matter of using the skills we learned way back in the linearity section to undo addition and division and subtraction, and we eventually end up with that *x* = -6.

But sometimes the problems will not be as straightforward and will make you do some work before you can bust out the log. Take this one: (1/8)(6*x* + 2) = 4(6*x* + 12).

Again, I have variables stuck in exponents on both sides, so I'd like to undo the exponentials on both sides. But because the bases are different, taking, for example, the log base 4 of both sides would only cancel out the right. On the left I would have a pretty messy log that I'd rather not deal with. So in order to have both bases go away - both exponentials go away - I need both bases to be the same.

Luckily 1/8 and 4 can both be rewritten as powers of 2. By remembering that negative exponents give us fractions, I can say that 1/8 is the same thing as 2-3. I can also say that 4 is 22, and now I have the base of my exponentials the same. I do have to use the power of a power property to multiply exponents together when they are raising each other, which leaves me with the equation 2(-18*x* - 6) = 2(12*x* + 24). Now that both bases are the same, this problem is just like the previous one. I can take the log base 2 of both sides, both bases magically disappear, and I simply get -18*x* - 6 = 12*x* + 24. Again, we solve with the inverse operations that we've hopefully known for a while now, undoing subtraction with addition, undoing addition with subtraction, undoing multiplying with dividing, and we find out that *x* must have been -1.

To review, exponential equations often appear when dealing with interest and money and can be solved by using logarithms.

When you have two exponentials that don't have the same base, try to rewrite them as the same number so that you can use the same log to cancel both of them out.

We can also solve exponential equations with exponentials on both sides of the equation, but remember that when the bases of those exponentials aren't the same, you'll have to first make them the same before you can undo them with a log.

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

Create
your account

Already a member? Log In

BackDid you know… We have over 95 college courses that prepare you to earn credit by exam that is accepted by over 2,000 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 | 94 lessons | 11 flashcard sets

- Computer Science 108: Introduction to Networking
- Psychology 316: Advanced Social Psychology
- Hiring & Developing Employees
- Accounting 305: Auditing & Assurance Services
- MTEL Physical Education (22): Study Guide & Test Prep
- The Transmission Control Protocol/Internet Protocol Model
- Computer Networking Fundamentals
- Network Topologies & Ethernet Standards
- TCP/IP Mail Services & Network Troubleshooting
- Crimes Against Children & the Elderly
- Study.com CLEP Scholarship for Military Members
- Study.com Scholarship for Texas Students & Prospective Teachers
- Study.com Scholarship for Florida Students & Prospective Teachers
- What are TExMaT Exams?
- What is the Florida Teacher Certification Examination (FTCE)?
- Study.com TExES Scholarship: Application Form & Information
- Study.com FTCE Scholarship: Application Form & Information

- Forensic Laboratories: Description & Services
- Using the Eisenhower Decision Matrix to Prioritize Tasks
- Arson: Definition, Motivation & Types
- How to Draft a Job Ad that Promotes Inclusion
- Using Manipulatives to Solve Probability Problems
- Overcoming Cognitive Biases & Judgment Errors in Decision Making
- Gathering Background Information on Students with Autism Spectrum Disorder
- How Social Media Affects Behavior: Online & Offline
- Quiz & Worksheet - Statutes in Law
- Quiz & Worksheet - Teaching Factoring
- Quiz & Worksheet - Analyzing The Other Two
- Quiz & Worksheet - Forensics in the Modern World
- Quiz & Worksheet - Fingerprints Attributes
- International Law & Global Issues Flashcards
- Foreign Policy, Defense Policy & Government Flashcards

- GRE Prep: Help and Review
- Accuplacer Arithmetic Test: Practice & Study Guide
- The Civil War and Reconstruction: Help and Review
- Common Core ELA Grade 7 - Literature: Standards
- Animal Behavior Study Guide
- HiSET: Energy & Heat
- MTTC Psychology: Introduction to Research Methods
- Quiz & Worksheet - Types of Thermodynamic Processes
- Quiz & Worksheet - Sexual Harassment Effects
- Quiz & Worksheet - Archaism in Literature
- Quiz & Worksheet - Circle Graph
- Quiz & Worksheet - Counterpoint in the Baroque Period

- Octagon in Geometry: Definition, Properties & Formula
- Dissection Techniques & Alternatives
- AP Calculus Exam Calculator: What's Allowed?
- John F. Kennedy and the Vietnam War: Learning Objectives & Activities
- Star Spangled Banner Lesson Plan
- How Does DSST Scoring Work?
- What is the GACE Test?
- Found Poetry Lesson Plan
- GRE Math Test & Study Guide
- Mole Day Project Ideas
- Scientific Revolution Lesson Plan
- Light for Kids: Activities & Experiments

Browse by subject