 Welcome back to our lecture series, Math 1050, College Algebra for Students at Southern Utah University. As usual, I'll be your professor today, Dr. Andrew Misildine. In lecture 37, we are beginning, essentially our final unit, unit six about Exponential and Logarithmic Functions. We'll see later on in this series that our last unit, seven, which we'll entitle pre-calculus, we're going to review most of the things we've seen in this series. Unit six offers the last new content for this lecture series on College Algebra. Like I said, this unit is going to be about Exponential and Logarithmic Functions. Instead of just jumping into the deep end of the pool here, in this chapter, I actually wanted to motivate the Exponentials and Logarithmic Functions a little bit more than I have in other topics. In this lecture, we're going to focus on the idea of money. That is, let's talk a little bit about some financial mathematics and how these naturally lead to questions about Exponential Functions. Say we open a new savings account, I want to track the growth of the account. Suppose that we have some initial deposit, we're going to call it P, and this is referred to as the principal of our investment. Now, just remind ourselves that earlier in this series, when we talked about linear functions, we introduced the idea of simple interest. What I want to do right now is develop the notion of compounded interest, which while simple interest leads to linear functions, compounded interest leads very naturally to Exponential Functions. So we have this principal P, which is the initial deposit we made into the savings account. And then there's going to be an interest rate, and which for the lack of a better name, I'm going to call it I% right now. Now, I'm just using I here as a variable. This is not the imaginary unit. That would be kind of weird to take I% if that was an imaginary number. I just mean, there's some interest rate. We're going to call it I right now, just for the sake of label. And this is going to be an interest unit, an interest rate per unit of time, right? So this might be like, oh, you make 5% annually or 1% monthly, something like that. So it's typically in finance though, when you talk about time, time is usually measured in years. And so we can talk about interest rate per year annually, something like that. So after the time period, whatever that turns out to be has elapsed, we are going to gain P times I dollars, right? And then the total balance would look like the following. So after one unit of time, we would get that A equals P plus PI. So as you dissect this, what does it mean? Well, you're going to get back your original principle, right? If you don't get back your original principle with your investment, that's not a very good investment. We should get our principle back. But then in addition to principle, we collect interest on that principle. So you're going to get a percentage of the original investment P added to that. Well, as a mathematician, I cannot help myself. I see an expression like P plus PI. I have to factor it. And this turns out to be P times one plus I, right? And this right here is essentially just the simple interest formula we saw previously. That the amount will equal the principle times one plus I, which although we didn't call it I at the time, we called it R, the rate. And then we also put a unit of time. So as multiple units of time elapsed, we get this simple interest rate. Well, for this consideration, we're doing time equals one at the moment, all right? And so that's why you don't see a T and we're just using I instead of R for a moment for interest rate, okay? So that's how much we'd have after one unit of time. And I do want to kind of keep track of that. When we have one unit of time, that's how much money we'd have. Well, what if we don't withdraw anything from the account? If we wait another time unit, our balance will grow from the interest of the initial principle and from the interest gathered on the principle itself. So this is the idea behind compounded interest. What if the interest starts collecting interest? So after a second unit of time, which we'll just say after two years have elapsed, then we're gonna have the amount we had previously. So we're gonna have our principle back again. We're gonna get back the interest we gathered off the principle. So again, factoring this, we have one plus I again. So we get the amount back we had previously, but then we're gonna collect interest on this, P times one plus I times I. Basically how I want you to think about it, it's the second unit of time. It's as if we were doing the first unit of time, but we had a bigger principle this time. But again, as a mathematician, I can't help myself. I notice there's a common divisor. P is common to both, but we also have a one plus I that's common to both. If I factor those out, this will give me that the amount is equal to P times one plus I. That's what I factored out. But then what's left behind when you factor these things out, that'd give me a one, and this one would give me an I. So you get P times one plus I times one plus I. We can factor that even better. We get the amount is equal to the principle times one plus I squared. And so after two units of time, the amount in the account would be P times one plus I squared, all right? What if we were to do this for another year? What if we wait another unit of time? Well, for this third time unit, how much would we have in the account? Well, we're gonna have the same amount we had the previous year, but then we collected interest on this entire amount, not just interest on the principle P, but we got interest off of every amount that's in there. So interest is collecting interest, so the interest is compounding. So we're gonna take the amount from the previous year, P times one plus I squared, and then we're gonna times that by I, for which then you should notice there's a common divisor of P times one plus I squared. When we factor that out, we end up with P times one plus I squared, but then the first group leaves behind a one, the second group leaves behind an I, and when you factor that further, you get P times one plus I cubed, all right? And we start to see a pattern that emerges. The next time around, when we look at, okay, four units of time have elapsed now, we still haven't made any withdrawals here, then the amount is gonna equal P times one plus I cubed plus the amount from previously, plus the interest we collect, P times one plus I cubed times I, and when you factor that, you're gonna get the amount is equal to P times one plus I to the fourth. And so then we have to ask, well, what's the pattern here if we continue down in this trajectory, right? If we continue down this trajectory, what happens? What happens when we equal the nth moment in time, right? Well, and again, I'm just gonna actually leave it just unspecified as T, like what if we just come to the teeth moment, right? We're not gonna specify what it is. Well, looking at our pattern we've seen so far, notice we see the following. The amount is equal to P times one plus I, and I'm gonna write this to the first power. T squared, right? Is gonna give us this one right here, and T squared, I wanna talk about T equals two. You'll get A equals P times one plus I squared. When T equals three, you get A equals P times one plus I cubed. For the fourth time, you're gonna get A equals P plus I to the fourth. This seems to suggest where we're going here. If we don't specify the time unit, it looks like the amount is gonna equal P times one plus I to the teeth power. So whatever T turns out to be, you're just gonna take an exponent with that regard. I do wanna add one more to our list. What about T equals zero, right? What about this start of the experiment? At the start of the experiment, we should only have the principle we've made no interest yet, but that actually agrees with our formula, oops, because if we take a number to the zero of power, that's just equal to one, right? So A to the zero is equal to one with an exponent law we've seen previously. So even when we take T equals zero, that seems to match up. And so you see that with this compounded interest scenario we see right here, what is happening is that this growth, the growth of this interest is actually growing with respect to the exponent here, T. All right, so now let's make, let's make another change to this conversation here. So instead of the interest rate I, I'm going to make a substitution where we're gonna say I is equal to R divided by N. So what do these numbers mean here? R is gonna be our annual percentage rate or sometimes called APR for short. So this would be the interest rate for the entire year. And the symbol N, I'm gonna introduce now, we're gonna call this the number of compounds per year. So what I mean by a compound. So when it comes to finance, although interest rates are usually always given as annual percentage rate, APR, when you collect interest could happen more frequently. So for example, we could have, if you had an annual interest rate, of course, that would suggest that N equals one. You could collect interest on an investment semi-annually in which case that means the number of compounds is two. If you had quarterly interest, whoops, quarterly interest, we'd be saying that your compound is four. For your usual consumer, if you have like a savings account or something, your interest is probably monthly. So you get 12 compounds per year, okay? And then you could have like bi-weekly, weekly, whatever. You know, the number of compounds tells us how frequently we are collecting interest here. And so then this number right here, I equals R over N, this is gonna tell you the percentage per compound, right? So what's the interest rate per compound? So if you had, for example, 12% annual interest, then if you got interest monthly, right? Then that actually would be, your monthly interest rate would be 1%. You just take 12 divided by 12. And I should apologize that the examples we're gonna be using in these videos are examples that came from mathematical textbooks we've written prior to 2008. And so they might not, you know, for the year 2020 when this video was recorded, they don't look like realistic interest rates. Now, we're not doing this from a financial investment. I'm not giving any strategy on how you should invest. I'm just gonna teach you the mathematics of these equations. So the actual interest rates are really somewhat irrelevant, but I don't want anyone to get depressed by these things that we'll see in examples in just a moment. But when you take this formula we see right here and combined it with this one right here, we see the following formula. We're gonna get A of t. So we're treating this as a function with respect to time. You're gonna get p times one plus r over n times, and I wanna make my r look more like an r here, r over n times nt. And this right here is referred to as our compounded interest formula. And so again, how do you interpret this formula? p is the principle, this is the initial investment. r is gonna be the APR, the annual percentage rate. n is the number of compounds per year. So we take the annual rate and divide it by how frequently interest is collected. That'll then give you the interest per compound. And then we're gonna have up here nt where t is gonna be measuring years. In finance problems time will always be measured in years. Growth in investments is often slow going years as an appropriate measurement of time. Now, if you take t and you times it by n, n is the number of compounds per year. So nt is gonna be the number of compounds that have occurred. So for example, if a usual car loan is done for a five year time period, after a 12 months per year, that means for your typical car loan, there'll be 60 installments. That is 60 payments you would pay because you take 12 times five years there. Okay, so this is our compounded interest formula. It's a beefed up version of the simple interest formula we saw before and it more accurately measures how usual interest problems are computed in finance. Much more realistic. Simple interest, basically mom and dad might even give you a simple interest loan but that's about it nowadays. So let's see an example of how one can do some computations using this compounded interest formula. Suppose that a credit union pays an interest rate of 8% per annum, that just means per year, right? And annum here is just a Latin word for year, kind of like why we say annual. 8% APR compounded quarterly on a certain savings plan. If $1,000 is deposited in such a plan and the interest is left to accumulate, how much is the account after one year? So we're gonna use our compounded interest formula. We have to compute the amount. That's what they want us to know. What is, how much is in the account, right? That's the amount. This will equal P times one plus R over N times NT. So we need to know all of these variables here. So principle is the initial investment which we see that was $1,000, okay? Then the one is a constant. We need the rate. This is the annual percentage rate and we see that's gonna be 8% per annum per year. So when it comes to these formulas, don't put in an 8, you're gonna put 0.08 because it's 8%, we should write it as a decimal in the formula. N is gonna be the number of compounds which we see it's compounded quarterly. Like we mentioned in the previous slide that means N should equal four. And then the last thing that's determined is T equals years, right? And so we want to invest for one year, so T equals one. So then we just have to compute A by plugging in these values. We're gonna take 1,000 times one plus 0.08 divided by four raised to the four times one. So essentially in a year, four quarters will have occurred. So computing some of these things, we see for example that four goes into 8%, well, 0.02 times. That is the quarterly interest rate would be 2%. Four times one is four. You're gonna end up with 1,000 times, well, now you're gonna get one plus 0.02. That's gonna be 1.02, right? So we see this, we see there's gonna be a 2% growth per month of, excuse me, per quarter and there's gonna be four quarters. So we have to compute this. Now be aware, we could compute this by hand. Taking the fourth power just means 1.02 times 1.02 times 1.02 times 1.02. But oftentimes these calculations are gonna be very messy at best and darn near impossible all the time. So these are type of calculations. You're gonna wanna use a calculator, preferably a scientific calculator that has both exponential and logarithmic buttons. We'll talk about logarithms later on in this unit here. So don't be a hero, use a calculator on this one. Now you wanna try to keep as many decimals in the expansion as possible on your calculator. So if you take 1.02 and take it to the fourth power, your typical scientific calculator will give you 1.082431, excuse me, 3216, like so. And so notice that you have 1, 2, 3, 4, 5, 6, 7, 8. You have eight decimal places, that's very good. We won't have a lot of decimal places because even these small decimal places when you start raising them to large powers potentially could have an impact on the final result. Cause after all, when you time something by 1,000, that moves the decimal place by three locations, right? So we wanna make sure that we have lots of decimal places. As these financial problems in this series will focus around US dollars. So our official currency in this conversation will be USD. Whenever you're working with US dollars, we always write round the final answer to the nearest penny that is to the nearest 100th, that's two decimal places. But to make sure that our round is not incorrect, we don't wanna round off by a penny or something like that. People get mad when you round incorrectly their money cause that often means they lose more. So we will, we're gonna have lots of decimal. So eight decimal places is fairly safe since we're gonna be multiplying things by 1,000. But if we invested like a million dollars, we need a lot more decimals to make sure that interest is calculated correctly, okay? Which admittedly on this one, I do believe this isn't, I didn't round this one. If you actually take 1.02 to the fourth, you get exactly this decimal expansion. So times in it by 1,000, like I said, that's just gonna move the decimal over by three locations. So we go 1, 2, 3. And so we end up with 1,082 and 43 cents. So the next decimal would be a two, I would round that down. And so this is how much the account should be worth. So notice you get back the original thousand dollars. The interest that we would have collected over this year is gonna be $82 and 43 cents. And that's pretty nice for not even, I didn't do anything, just I let the bank invest my money and it gave me about $83 back in interest. Well, that's of course with an 8% interest rate, which again, that's difficult to find in the year 2020. Let's look at another example here. So what if, let's see, what if annual interest rate compounded annually, excuse me, what annual interest rate compounded annually should you seek if you wanna double your investment in five years? So if you expect it to double, right? What does the interest rate have to be? Assuming you could actually shop around for such a thing. By all means, I would love to find a bank that would double my investment in five years. But let's just kinda keep things simple here. Just treat it as a mathematical exercise. What can we do? Well, some things we can unravel here. Compounded annually, that would say that the number of compounds is equal to one, okay? Five years, that's the timeframe, five here. This, it's asking what rate, right? So we actually don't know what the rate is. That's what we're trying to solve for. And then we want the investment to double, okay? So we don't necessarily know what the principle is either, but we can say we want the amount to be 2P, whatever the principle is, we wanna double it, okay? So looking at our formula, A equals P times one plus R over N to the NT. Let's plug in what we know. So A will become a 2P. P, we don't know what it is. That seems a little bit concerning to us, but actually it turns out we don't need to know what it is. We're gonna go one plus the rate, we don't know. That's actually what we need to solve for. The compounded annually is one. And then we get one times five. So what we can do is the following, we get 2P equals P times one plus R to the fifth power. Great, this is the equation we have to solve. And it seems a little bit concerning because we have two variables, right? Principle and rate. Rates what we need to solve for both our principle. Well, the good news is the principle doesn't actually matter because it's a fair assessment, a fair assumption to say that the principle is not zero. If you want your investment to double, you need something to double. If I deposit zero dollars in the bank, then voila, it doubled instantaneously because twice to zero is zero, right? So we have to invest something, whether that's a dollar, a million dollars, five dollars. We have to invest something if we expected to double. So we can assume that P is not zero, which means we can actually divide both sides by P. And then we end up with the equation two equals one plus R to the fifth. So the cool thing about this one is that the time it takes to double your investment doesn't matter. Whether you invest $1, $100 or $10,000, the time it takes to double is the same. And so this is actually a good advice when it comes to investments, right? That if you're trying to save up for retirement, it's a good idea to start putting lots. Even if it's a small amount, you want to put money into your account even when you're young because its ability to grow interest is irrelevant of how much is in there, but the more that's in there, the more interest you'll collect. It'll double irregardless of how much principle you have. So then we solving for R right here, we're going to take the fifth root of both sides to get rid of the power of five. So we end up with one plus R equals the fifth root of two. Now this is another smart suggestion when you're doing your problems on a calculator that even though I could approximate the fifth root of two right now, I want to keep the exact answer exact until the very end. Because if I start approximating things too early, then the rounding errors might get out of hand. So then to solve for R, we're going to subtract one. And so the exact answer will be the fifth root of two minus one for which then I'm going to put that in my calculator and I'm going to end up with, well, the fifth root of two is 1.148698. So if you subtract one from me, you're going to get 0.148698. And so let's round this to two decimal places. Our interest rate R should be approximately 14.87% if we want to double our investment in five years.