How to Solve Exponential Equations

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  • 0:06 Exponential Equations…
  • 1:03 How to Solve an…
  • 2:25 Using the Change of…
  • 3:13 Solving With Similar Bases
  • 3:51 Solving With Different Bases
  • 5:41 Lesson Summary
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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!

Exponential Equations in the Real World

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

Calculating how fast your money in the bank will grow is an example of an exponential equation
Exponential equations in the real world

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.

How to Solve an Exponential Equation

Substituting the variables in the correct places in the equation would this time give me 4,000=500e0.06(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=e0.06(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.

Using the Change of Base Formula

Use the change of base formula to undo exponentials that are not commonly found on a calculator
Change of base formula

As a quick note, when we use logs to undo exponentials that aren't the common ones on our calculator, say, undoing 8x = 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 8x, 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.

Solving With Similar Bases

We can also solve exponential equations that have exponentials on both sides, like this: 5(2x - 1) = 5(3x + 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 2x - 1 = 3x + 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.

Solving With Different Bases

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)(6x + 2) = 4(6x + 12).

Make bases the same so it is easier to solve problems with different bases
Solving with different bases

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