In the last video, I hopefully showed you that if I borrowed P dollars, and I borrow it for a year, and you were to charge me an interest rate of r, or you could say 10r%, then at the end-- and we were to compound continuously. So we compound every zillionth of a second, but we compound it a trillion times, however many of those intervals are in a year, that at the end of a year, I would owe you P times e to the r dollars. Fair enough. Now, what happens if I borrow it for two years? Well, after one year, we already said that I would owe you P times e to the r dollars, right? And then after two years, what happens? Well, this becomes the new principal. You can kind of view it as like I borrowed this much, then I owed this much after a year, and so this is the new principal, so I can reborrow this, right? So if I reborrow this, this becomes the new P. So that becomes the new P, so I could write Pe to the r, and it's going to-- and that new principal is going to compound for another year. So e to the r. So that equals Pe to the 2r. And similarly, this is now my new principal. If I were to borrow it for another year, it becomes Pe to the 3r. So, in general, if I borrow P dollars, that's my initial principal, I borrow it at a rate of r and I borrow it for two years, the amount that I owe after two years is Pe to the rt. And once you know this, you are ready to become your local banker and to lend people money continuously. And let me just do a couple of examples because I think it might be a little confusing in the abstract, but with some numbers it might all clear up. OK, so let's say I borrow $1,000. Let's say that the interest rate is 25%. That's the annual interest rate. Rate is equal to 25%, which is the same thing as 0.25. And let's say I were to borrow it for three years. So t is equal to three years. And we're going to continuously compound this interest. So our formula says that the amount that I'll owe at the end of this is how much I borrowed,$1,000 times e, to my interest rate power, 0.25, times the number of years, times t. And so let's-- oh, sorry, that's 3, right? So that equals 1,000e to the 0.75 power, and let me calculate what that is in Excel. And just so you know, I don't know if you're familiar with Excel. In Excel, the e to the power-- so I wrote that it's a 1,000 times-- e to a power in Excel is exp. So that's e to some power; in this case, it's 0.75. So I get my answer. I don't know. I think it fell off the bottom of the screen. There it is right here. That's my answer. Zoom in a little bit because I think you might have trouble reading it because it kind of shrinks it when I go to YouTube. $2,117. It equals$2,117, and that's what you would owe me at the end of three years. This is actually the power of compounding interest. A lot of people, you know, when you hear 10% interest rate or even a 25% interest rate, that no one really makes a big deal about it. But when you compound it, and especially when you compound it continuously, it can very quickly turn into very, very large numbers. But let's do another example, and this might be another kind of more complicated example, or something that you might actually see in a textbook. Let's say that I borrow $50. I borrow$50, and let's say it's continuously compounded at some rate r, and let's say it is continuously compounded for 10 years. At the end of 10 years, I owe $500. What was the rate at which it was compounded? So once again, we can use the same formula. We could say, well, if my original principle is$50-- so it's going to be $50 times e to the rate. We don't know the rate, but we know that t is equal to 10 years, so its 10r. That equals my final payment or how much I owe once all the interest and the principal has compounded. It's equal to$500. So we can divide both sides by 50. You get e to the 10r is equal to 10. And then how do we solve that? Well, we could take the log base e of both sides. Hopefully, you might want to review the logarithm, but log base e-- e is just a number, if you ever get confused-- is equal to log base e of 10. And log base e on your calculator is often written the natural log. And they called it the natural log because I'll show you e in a hundred different-- not a hundred, but in many different applications. It shows up all over nature, and I think that's why it's called the natural log. Anyway, let me see if I can figure out what Excel's natural log function is. So I need to figure out the natural log, log base e of 10. Equals LN 10. Oh, there we go. There it is right there: 2.3. So first of all, if I say log base e of e to the 10r, that's like saying e to what power is equal to e to the 10r? So this is the same thing as just 10r. Why is that? Because remember, logarithm is an exponent, so this is saying e to the 10r is equal to e to the 10r. Review my logarithm videos if that's a little confusing. I know it's a little confusing at first. And then we just figured out that log base e, so e to the what power is 10, is 2.-- what was the number? 2.30. And now-- oh, this isn't 10 to the r. This is 10r, right? And so we want to figure out what r is. We divide both sides by 10. We get r is equal to 0.23, or 23%. So essentially, if I continuously compound at an annual rate of 23%, after 10 years, I'll essentially owe 10 times the money. So that's something good to keep it mind. Anyway, I'll leave you there, and I really encourage you to go back a couple of videos, rewatch them, play with the numbers. Prove to yourself that that limit exists. Take that limit that we showed in the beginning, the limit as n approaches infinity of 1 over 1 plus n to the n. All you have to do to prove this is just put in larger and larger numbers for n. And, of course, whatever number you put in there, you have to put over here. You can't put a million here and a trillion there. You have to put a trillion and a trillion, or a million and a million. And you'll see that it converges to e. And rewatch the videos and make sure you get kind of an intuitive understanding of everything we did. And then this formula, which most people, frankly, just memorize, this Pe to the rt, will make a lot of sense to you, and you will have a permanent neuron for it the rest of your life. Anyway, I'll see you in the next video.