For more information, please see full course syllabus of Math Analysis

For more information, please see full course syllabus of Math Analysis

### Exponential Functions

- An
*exponential function*is a function of the form

where x is any real number and a is a real number such that a ≠ 1 and a > 0. We call a thef(x) = a ^{x},*base*. [The base is the thing being raised to some exponent.] - From the previous lesson,
*Understanding Exponents*, we can compute the value of a given base raised to any exponent. In practice, we can find these expressions (or a very good approximation) by using a calculator. Any scientific or graphing calculator can do such calculations. - Exponential functions grow really,
__really__, REALLY fast. There's a fictional story in the lesson to get across just how incredibly fast these things grow. Multiplying by a factor repeatedly can get to extremely large values in a short period of time. -
*Compound interest*is a form of interest on an investment where the interest gained off the*principal*(the amount of money initially invested) also gains interest. Thus, compound interest returns more and more money the longer the investment is left in. - We can describe the amount of money, A, in a compound interest account with an exponential function:
A(t) = P ⎛

⎝1 + r n

⎞

⎠

, - P is the
*principal*in the account-the amount originally placed in the account, - r is the annual
*rate*of interest (given as a decimal: 6% ⇒ 0.06), - n is the number of times a year that the interest compounds (n=1⇒ yearly; n=4⇒ quarterly; n=12 ⇒ monthly; n=365 ⇒ daily),
- t is the number of years elapsed.

- P is the
- In the above equation, we can see that the more often the interest compounds, the more money we make. This motivates us to create a new number: the
*natural base*, e. (See the video for more information on how we find e.) The number e goes on forever:

Like π, the number e is an irrational number: its decimal expansion continues forever, never repeating. Also like π, the number e is deeply connected to some fundamental things in math and the nature of the universe. Just as π is fundamentally connected to how circles work, the number e is fundamentally connected to how things that are continuously growing work.e = 2. 718 281 828 459 045 235 360 … - One application of e is to see how an account would grow if it was being compounded every single instant- being
*continuously compounded*:A(t) = Pe ^{rt},- P is the principal (starting amount),
- r is the annual rate of interest (given as a decimal),
- t is the number of years elapsed.

- We can see exponential
*decay*if 0 < a < 1 in f(x) = a^{x}. The exponential function will quickly become very small as it repeatedly "loses" its value because of the fraction (a) being compounded over and over.

### Exponential Functions

Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.

- Intro 0:00
- Introduction 0:05
- Definition of an Exponential Function 0:48
- Definition of the Base
- Restrictions on the Base
- Computing Exponential Functions 2:29
- Harder Computations
- When to Use a Calculator
- Graphing Exponential Functions: a>1 6:02
- Three Examples
- What to Notice on the Graph
- A Story 8:27
- Story Diagram
- Increasing Exponentials
- Story Morals
- Application: Compound Interest 15:15
- Compounding Year after Year
- Function for Compounding Interest
- A Special Number: e 20:55
- Expression for e
- Where e stabilizes
- Application: Continuously Compounded Interest 24:07
- Equation for Continuous Compounding
- Exponential Decay 0<a<1 25:50
- Three Examples
- Why they 'lose' value
- Example 1 27:47
- Example 2 33:11
- Example 3 36:34
- Example 4 41:28

### Math Analysis Online

### Transcription: Exponential Functions

*Hi--welcome back to Educator.com.*0000

*Today, we are going to talk about exponential functions.*0002

*Previously, we spent quite a while looking at functions that are based around a variable raised to a number--*0005

*things like x ^{2} or x^{47}; this is basically the idea of all of those polynomials we have worked with for so long.*0010

*But what if we took that idea and flipped it?*0017

*We could consider functions that are a number raised to a variable, things like 2 ^{x} or 47^{t},*0019

*where we have some base number that has a variable as its exponent.*0027

*We call functions of this form exponential functions, and we will explore them in this lesson.*0032

*Now, make sure that you have a strong grasp on how exponents work before watching this.*0036

*If you need a refresher on how exponents work, check out the previous lesson, Understanding Exponents, to get a good grounding in how exponents work.*0040

*All right, an exponential function is a function in the form f(x) = a ^{x},*0048

*where x is any real number, and a is a real number such that a is not equal to 1, and a is greater than 0.*0054

*We call a the base: base is just the name for the thing that is being raised to some exponent.*0060

*So, whatever is being exponentiated--whatever is going through this process of having an exponent--that is called the base,*0066

*because it forms the base, because it is below the exponent.*0072

*We might wonder why there are all of these restrictions on what a can be; well, there are good reasons for each one.*0076

*If a equals 1, we just have this boring constant function, because we would have 1 ^{x}, which is just equal to 1 all of the time.*0081

*So, something that is just equal to 1 all of the time is not really interesting, and it is not really going to be an exponential function.*0091

*So, we are not going to consider that case.*0097

*If a equals 0, the function wouldn't be defined for negative values of x.*0100

*If we try to consider what 0 ^{-1} is, well, then we would get 1/0, but we can't do that--we can't divide by 0.*0104

*So, that is not allowed, so that means a = 0--once again, we are not going to allow that one.*0111

*And if we had a < 0, then the function wouldn't be defined for various x-values, like x ^{1/2}.*0116

*For example, if we had -4, and we raised that to the 1/2, well, we know that raising it to the 1/2 is the same thing as taking the square root.*0121

*So, the square root of -4...we can't take the square root of a negative, because that produces imaginary numbers;*0129

*and we are only dealing with real numbers--we are not dealing with the complex numbers right now.*0134

*So, we are going to have to ban anything that is less than 0.*0137

*And that is why we have this restrictions: our base has to be greater than 0, and is not allowed to be 1,*0140

*because otherwise things break down for the exponential function.*0145

*All right, notice that, from the previous lesson, we can compute the value of a given base raised to any exponent.*0149

*We know how exponents work when they are a little more complex (not complex numbers, but just more interesting).*0154

*And so, we can raise things like...4 ^{3/2} = (√4)^{3},*0160

*which would be equal to...√4 is 2; 2 ^{3}, 2 times 2 times 2, gives us 8; great.*0165

*If we had 7 ^{-2}, well, then that would become 1/7, because we have the negative, so the negative flips it to (1/7)^{2}.*0171

*So, 1 squared is 1; 7 squared is 49; we get 1/49.*0179

*So, we can do these things that are a little more difficult than just straight positive integers.*0184

*But we might still find some calculations difficult, like if we had 1.7 ^{6.2}--that would probably be pretty hard to do.*0190

*Or (√2) ^{π}--these would be really difficult for us to do.*0196

*So, how do we do them? In practice, we just find these expressions, or a very good approximation, by using a calculator.*0200

*We can end up getting as many digits in our decimal expansion as we want.*0206

*We can just find as many as we need for whatever our application is--whatever the problem asks for--by just using a calculator.*0214

*Any scientific or graphing calculator can do these sorts of calculations.*0220

*There will be some little button that will say x ^{y}, or some sort of _ to the _--some way to raise to some other thing--something random.*0223

*They might have a carat, which says...if I have 3 ^{6} (not with an a--I accidentally drew that in...oh, I drew it in again),*0235

*then that would be equivalent to us saying 3 ^{6}.*0250

*The carat is saying "go up," so the calculator would interpret 3 ^{6} as 3^{6}.*0253

*There are various ways, depending on if you are using a scientific calculator,*0259

*or if you are using a graphing calculator, to put these things into a calculator and get a number out.*0261

*So, we are able to figure these things out, just by being able to say "use a calculator."*0266

*Now, from a mathematical point of view, that is a terrible statement.*0270

*We don't want to say, "We can deal with this because we have calculators!" because how did you figure it out before you had calculators?*0273

*Calculators didn't just spawn into existence and give us the answers.*0279

*We can't rely on our calculators to do our thinking for us; we have to be able to understand what is going on.*0283

*Otherwise, we don't really have a clue how it works.*0287

*But as you will see as you get into more advanced math classes, there are methods to figure out these values.*0290

*There are ways to do this by hand, because there are various algorithms that give us step-by-step ways to get a few decimals at a time.*0296

*Now, doing it by hand is long, slow, and tedious; it would be hard to get this sort of thing, just because it would be so much calculation to do.*0303

*We could do it, but that is what calculators are for; they are to do lots of calculations very quickly.*0310

*They are to help us get through tedious arithmetic.*0316

*So, since these sorts of calculations take all of this arithmetic, we designed calculators that can do this method for us.*0319

*And that is why we can appeal to a calculator--not because the calculator knows more than us,*0326

*but because, at some point, humans figured out a method to get as many decimals as we wanted to;*0331

*and then, we just built a machine that is able to go through it quickly and rapidly,*0336

*so we can get to the thing that we want to look at, which is more interesting, using this.*0340

*The calculator is a tool; but it is important to realize that we are not just relying on it because it has the knowledge.*0345

*We are relying on it because, at some point, we built it and put these methods into it.*0350

*And if you keep going in mathematics, you will eventually see that these are where the methods come from--there is some pretty interesting stuff in calculus.*0354

*All right, now, if we can evaluate at any place--if we can compute what these values of exponential functions are--*0361

*then we can make a graph, because we can plot as many points as we want; we can draw a smooth curve.*0368

*So, let's look at some graphs where the base is greater than 1.*0372

*If we have 2 ^{x}, that would be the one in red; 5^{x} is the one in blue, and 10^{x} is the one in green.*0375

*Now, notice: 2 ^{x}, 5^{x}, and 10^{x}--all of these end up going through 1, right here,*0384

*because what is happening there is that 2 ^{0}, 5^{0}, 10^{0}...anything raised to the 0--they all end up being 1.*0392

*Remember, that is one of the basic properties of exponents.*0401

*If you raise something to the 0, it just becomes 1; so that is why we see all of them going through the same point.*0403

*And notice that they get very large very quickly.*0409

*By the time 2 is to the fourth, it is already off; and 10 is off by the time it gets to the 1.*0411

*10 ^{x} grows very quickly, because it is multiplying by 10, each step it goes forward.*0417

*Notice also: as we go far to the left, it shrinks very quickly.*0421

*Let's consider 10 ^{-3}; 10^{-3} would be the same thing as 1/10^{3}, which would be 1/1000.*0425

*That is why we end up seeing that this green line is so low.*0436

*It looks like it is almost touching the x-axis; it isn't quite--there is this thin sliver between it.*0440

*But it is being crushed down very, very quickly, because of this negative exponent effect,*0444

*where it gets flipped over, and then it has a really, really large denominator very quickly.*0448

*So, we see, as we go to the left side with these things, that it will crush down to 0.*0454

*And as we go to the right, it becomes very, very big.*0460

*We can change the viewing window, so that we can get a sense for just how big these things get.*0464

*And look at how big: we have gotten up to the size of 1000 by the time we are only out to 10.*0468

*And that is on 2 ^{x}; if we look at 10^{x}, 10^{x} has already hit 1000 at 10^{3}.*0475

*At x = 3, it has managed to hit 1000 as its height.*0483

*This stuff grows really quickly; this idea of massive growth is so central to the idea of exponential functions.*0488

*We are going to have a story: there is this story that often gets told with exponential functions,*0494

*because it is a great way to get people to understand just how big this stuff gets.*0499

*So, let's check it out: All right, long ago, in a far-off land, there was a mathematician who invented the game of chess.*0505

*The king of the land loved the game of chess so much that he offered the mathematician any reward that the mathematician desired.*0513

*The mathematician was clever, and told the king humbly, "Your Highness, I thank you;*0519

*all I ask for is a meager gift of rice, given day by day on a chessboard."*0523

*"Tomorrow, I would like a single grain of rice give on the first square;*0528

*on the next day, two grains of rice given on the second square; then on the following day, the third day,*0533

*four grains of rice; and so on and so forth, doubling the amount every day until all 64 squares are filled."*0541

*So, the mathematician is asking for the first square, doubled, doubled, doubled, doubled...*0550

*The mathematician drew the king a diagram to help make his request clear.*0556

*On the first day of his gift, he would end up having one grain of rice on the first square.*0559

*On the second day, there would be a total of two grains of rice (1 times 2 becomes 2).*0567

*On the third day, there would be a total of 4 grains of rice (2 times 2 becomes 4).*0572

*On the next day, there will be 8 (4 times 2 becomes 8), and then 16, and then 32, and so on and so on and so on,*0578

*going all the way out to the 64 ^{th} day, doubling each time we go forward a square on the board.*0586

*The king was delighted by the humble request and agreed to it immediately.*0593

*Grains of rice? You can't get a lot of grains of rice on a single chessboard; "It will be very easy," he thought.*0597

*He ordered that the mathematician would have his daily reward of rice delivered from the royal treasury every day.*0603

*A week later, the king marveled at how the mathematician had squandered his reward.*0609

*After all, he only had to send him 2 ^{6} = 64 grains of rice that day.*0612

*Notice: on the seventh day, we are at 2 ^{6}--let's see why that is.*0617

*On the first day, we have 1 grain; on the second day, we have 2 grains; on the third day, we have 4 grains.*0624

*On the fourth day, we have 8 grains; on the fifth day, we have 16 grains; on the sixth day, we have 32 grains.*0631

*And thus, on the seventh day, we have 64 grains.*0638

*So, notice: we can express this as 2 ^{0}, 2^{1}, 2^{2}, 2^{3}, 2^{4}, 2^{5}, and then finally 2^{6} on the seventh day.*0642

*Why is this? Because on the very first day, he just got one grain.*0655

*Every following day, it multiplies by 2--it doubles.*0659

*So, that means we multiply it by 2; so we count all of the days after the first day, which is why, on the seventh day, we see an exponent of 6.*0662

*So, in general, it is going to be 2 to the (number of day minus 1).*0671

*We will subtract one to figure out the grains on some number of day.*0678

*So now, we have an idea of how we can calculate this pretty quickly and be able to get these things figured out.*0683

*Another week later, on the fourteenth day, the king sent him 2 ^{13} (remember, it is the fourteenth day,*0689

*so we go back one, because it has been multiplied 13 times) grains of rice, which is 8,192 grains.*0694

*And 8,192 grains is just about a very large bowl of rice.*0702

*The king was still amazed at the fantastic deal he was getting.*0706

*But he was glad that the mathematician was at least seeing some small reward.*0709

*He loved the game of chess, after all; and if he ended up feeding the mathematician for a year, that was great.*0713

*It seemed like a wonderful deal; he was willing to give him palaces, jewels, and massive amounts of money.*0719

*He can give him a little bit of rice for the great game of chess.*0724

*At the end of the third week, on the twenty-first day, the king had to send the mathematician a full bag of rice,*0728

*because in the kingdom, a full bag of rice contained precisely 1 million grains.*0734

*So, on the 21 ^{st} day, we have 2^{20} grains of rice, which ends up being 1 million, 48 thousand, 576 grains.*0739

*So we see here: after we jump these first six digits, we have one million plus grains of rice.*0747

*So, he has managed to get one million grains of rice (which is one bag of rice), plus an extra 48000 in change, in grains.*0753

*So, perhaps the mathematician was not as foolish as the king had first thought.*0761

*At the end of the fourth week, the king was starting to get worried.*0766

*On the twenty-eighth day, he had to send him more than 134 bags of rice, because 2 ^{27} is more than 134 million grains of rice.*0769

*So, we are starting to get to some pretty large amounts here.*0780

*Now, the royal treasury has a lot of rice; he is not worried--he has hundreds of thousands of bags of rice.*0782

*So, he is not too worried about it; but he sees that this is starting to grow quite a bit.*0788

*At that moment, the royal accountant bursts into the throne room and says, "Your Highness, I have grave news!*0793

*The mathematician will deplete the royal treasury! On the forty-first day alone, we would have to give one million bags of rice!"*0799

*because 2 ^{40} is here, so we have one million million grains of rice; so we have one million bags of rice,*0806

*which is more than the entirety that the treasury has in rice.*0816

*"And if we kept going--if we let it run all the way to the sixty-fourth day,*0819

*we would have to send him more rice than the total that the world has ever produced,*0823

*because we would be at 2 ^{63}, which would come out to be 9 trillion bags of rice."*0827

*Look; we have ones here; we have thousands here; we have millions here, billions here, trillions here, quadrillions here;*0835

*it would be 9 quintillion grains of rice; if we knock off these first ones, we see that we are still at 9 trillion bags of rice.*0842

*That is a lot of rice, and the world doesn't have that much by far.*0851

*So, the mathematician's greed has enraged the king, and the king immediately orders all shipments of rice stopped.*0856

*The mathematician is not getting any more rice, and the mathematician is to be executed!*0862

*Now, the mathematician, being a clever fellow, hears the soldiers coming down the road, and he escapes.*0866

*He fled the kingdom with the few bags of rice that he could manage to carry on his back,*0871

*and he had to find a new place to live, far, far away from the kingdom.*0875

*So, the moral of the story is twofold: first, don't be overly greedy--don't try to trick kings.*0880

* But more importantly than that, exponential functions grow really, really large in a short period of time.*0886

*They get big fast; even if they start at a seemingly very, very small, miniscule amount, they will grow massive if given enough time.*0895

*So, that is the real take-away here from this story.*0904

*Exponential functions get big; they can start small, but given some time, they get really, really big.*0907

*All right, let's see an application of this stuff.*0916

*When you put money in a bank, they will usually give you interest on your money.*0918

*For example, if you had an annual interest rate of 10% (annual just means yearly) on a $100 principal investment*0922

*(the amount that you put in the bank), the following year you would have that $100 still*0931

*(they don't take it away from you), plus $100 times 10%.*0935

*Now, 10% as a decimal is .10; so it is $100 times .10, so you would get that $100 that you originally started with, and you would have $10 in interest.*0940

*Great; but you could leave that interest in the account, and then your interest would also gain interest.*0948

*The interest is going to get interest on top of it; so we would say that the interest is compounded, because we are putting on thing on top of the other.*0954

*So, you have $110 in your bank account now, because you had $110 total at the end last time.*0961

*$110 gets hit by that 10% again; so you still have the $110, plus...now 10% of $110 is $11.*0968

*Notice that $11 is bigger than 10--your interest is growing.*0978

*Over time, you are getting more and more interest as you keep letting it stay in there.*0985

*You continue to gain larger and larger amounts with each interest.*0990

*Compound interest is a common and excellent way to invest money, because over time,*0993

*your interest gains interest, and gains interest, and gains interest.*0998

*And eventually, it can manage to get large enough to be even larger than the principal investment,*1002

*and be the thing that is really earning you money--the time that you have spent letting it compound.*1006

*We can describe the amount of money, A, in such an account with an exponential function: A(t) = P[(1 + r)/n] ^{nt}.*1011

*Let's unpack that: P is the principal in the account--the amount that is originally placed in the account.*1024

*So, in our example, that would be $100 put in; so our principal would be 100 in that last example.*1032

*r is the annual rate of interest, and we give that as a decimal: here is our r, right up here.*1038

*In the last one, that was 10%, so it was expressed as .10.*1045

*n is the number of times a year that the interest compounds; n is the number of times that we see compounding.*1050

*So, n = 1 would be yearly; n = 4 would be quarterly; n = 12 would be monthly; n = 365 would be daily.*1057

*In our last one, it compounded annually, every year; so it compounded just once a year, so n was equal to 1.*1066

*Notice that n also shows up up here; it is n times t.*1072

*And then finally, t is just the number of years that we have gone through; so it is times t.*1077

*So, let's understand why this is the case.*1084

*Well, if we looked at 10%, just on the $100, we would have $100 times 1 + 10%.*1086

*So, $100 times 1.1 equals $110.*1096

*Now, if we wanted to have this multiple times, well, the next time it is $110 times 1.1, again.*1102

*We would get another number out of it; and then, if we wanted to keep hitting it...*1109

*we can just think of it as (100 ^{1.1})^{t}, and that will just give us the amount of times*1112

*that the interest has hit, over and over and over--our principal times the 1*1119

*(because the bank lets you keep what you started with), plus the interest in decimal form,*1125

*all raised to the t--the number of years that have elapsed.*1131

*Now, what about that "divide by n" part?*1134

*Well, let's say that we compounded it twice a year; so they didn't just give you your interest*1136

*in a lump sum at the end of the year--they gave it to you in bits and pieces.*1140

*So, the first time it compounds, if they did it twice a year (let's say they did it semiannually, two times a year),*1144

*then it would be 1 (because they let you keep the amount of money), plus .1/2 (because they are going to do it twice in a year).*1151

*So, the first time in the year, we would get 100 times (1 + 0.05), 100 times 1.05.*1158

*The first time in the year it gets hit, you would get $105 out of that.*1169

*Now, they could do it again, and we would have $105 get hit with another one of 1.05, and then we could calculate that again.*1173

*And that would be the total amount that you would have over the year.*1181

*Now, notice: 105 times 1.05 is going to be a little bit extra, because we are getting that 5 times 1.05, in addition to what we would have ended up having.*1183

*We would have 105 times 1.05; 5.25...so we will end up getting 5.25 out of this.*1192

*So, we will have a total of 110 dollars and 25 cents.*1200

*So, by compounding twice in a year, we end up getting 25 cents more than we did by compounding just once in a year.*1208

*So, the more times we compound, we get more chances to earn interest on interest on interest.*1214

*1 + .1, divided by 2...it is going to happen twice in a year; so since it happens twice in a year,*1219

*we have to have the number of times that it is happening in a year, times the number of years.*1225

*So, at the twice-in-a-year scale, we would see 1.05 to the 2 times number of years, because it happens twice every year.*1230

*And this method continues the whole time; so that is why we have the divide by n, because the rate has to be split up that many times.*1240

*But then, it also has to get multiplied that many times extra, because it happens that many times extra in the year.*1246

*So, that is where we see this whole thing coming from.*1252

*Now, we noticed, over the course of doing that, that the more times it compounded, the better.*1255

*We earn more interest if it is calculated more often; the more often our account compounds,*1260

*the more interest we earn, because we have more chances to earn interest on top of interest.*1267

*So, we would prefer if it compounded as often as possible--every minute--every second--every instant--*1271

*if we had it happening continuously--absolutely constantly.*1278

*This idea of having it happen more and more often leads to the idea of the natural base, which we denote with the letter e.*1281

*The number e comes from evaluating 1 + 1/n to the n as n approaches infinity--as this becomes larger and larger--*1288

*because remember: the structure last time was 1 plus this rate, divided by n to the n times t.*1295

*So, if we forget about the times of the year that it is occurring, and forget about the rate, we get just down to (1 + 1/n) to the n.*1302

*So, we can see what happens as n goes out to infinity--what number does this become?*1310

*It does stabilize to a number, as you can see from this graph here.*1314

*So, by the time it has gotten to 40, it starts to look pretty stable; it has this asymptote that it is approaching, so it is starting to become pretty stable.*1317

*We can look at some numbers as we plug in various values of n.*1325

*At 1, we get 2; at 10, we have 2.594; at 100, we have 2.705; at 1000, 2.717; at 10000, 2.718; at 100000, it is still at 2.718.*1328

*And there are other decimals there; but we see that it ends up stabilizing.*1342

*As we put more and more decimal digits, as n becomes larger and larger and larger, we see more and more decimal digits that e is going towards.*1347

*e is stabilizing to a single value, and we see more and more of its digits, every time we keep going with this decimal expansion.*1356

*So, as we continue this pattern, e stabilizes to a single number.*1364

*Now, it doesn't stabilize to a single number where we have finished figuring it; we keep finding new decimals.*1367

*But we see that decimals we have found so far aren't going to change.*1372

*e is 2.718281828...and that decimal expansion will keep going forever.*1376

*Just like π, the number e is an irrational number; its decimal expansion continues forever, never repeating.*1382

*So, that decimal expansion just keeps going forever, just like π isn't 3.14 (it is 3.141...it just keeps going forever and ever and ever).*1389

*So, e is the same thing, where we can find many of the decimals, but we can't find all of the decimals, because it goes on infinitely long.*1399

*Now, also, just like π, the number e is deeply connected to some fundamental things in math and the nature of the universe.*1406

*e is connected to the very fabric of the way that the universe, and just things, work.*1414

*So, π is fundamentally connected to how circles work; circles show up a lot in nature, in the universe.*1418

*π is connected to circles, and e is connected to things that are continuously growing--*1423

*things that are always growing, that don't take this break between growth spurts, but that are just always, always, always growing.*1430

*e gives us things that are doing this continual growth; e has this deep connection;*1438

*and if you continue on in math, you will see e a lot (and also if you continue on in science).*1443

*One application of e is to see how an account would grow if it was being compounded every single instant.*1448

*That idea, that we are not just doing it every year; not just every day; not just every minute; not just every second;*1454

*but every single instant--that gives us P (our principal amount) times e ^{rt}.*1459

*The amount in our account is P times e ^{rt}; we can also just remember this as "Pert"; Pert is the mnemonic for remembering this.*1466

*P is the principal, or we can just think of it as the starting amount--however much we started with.*1474

*r is the annual rate of interest, and it can even be used for things that aren't just annual rates,*1478

*but r is the annual rate of interest; and remember: we give that as a decimal.*1483

*If we give it as a percent, things will not end up working out.*1487

*And t is the number of years elapsed.*1489

*Now, this above equation, this one right here--this "Pert" thing--this can be used for a wide variety of things*1493

*that grow or decay continuously--things that are constantly growing or constantly decaying.*1499

*You will see it show up a lot in math and science as you go further and further into it.*1503

*It is very, very important--this idea of some principal amount, times e to the rate times the amount of time elapsed.*1508

*You can use it for a lot of things; and while we will end up, in these next few examples, using some other things*1516

*than just e ^{rt} (with the exception of the examples that involve continuously compounded interest),*1521

*you can actually bend a lot of stuff that you have in exponents into using e.*1525

*So, it is easiest to end up just remembering this one, and then changing how you base your r around it.*1530

*Now, don't get too confused about that right now; we will see it more as we get into other things and logarithms,*1535

*and also just as you get further and further into math.*1540

*You will see how Pe ^{rt} is a really fundamental thing that gives us all of the stuff that is doing the growth.*1542

*Finally, exponential decay: so far, we have only seen exponential functions that grow as we go forward--*1550

*f(x) = a ^{x}, where our base, a, is greater than 1; so it gets bigger and bigger as we march forward.*1556

*But we can also see decay, if we look at 0 < a < 1--*1562

*if a is between 0 and 1--it is a fraction--it is smaller than 1.*1566

*Here are some examples: if we have 4/5 ^{x}, we see that one in red;*1571

*1/2 ^{x}--we see that one in blue; 1/10^{x}--we see that one in green.*1576

*Notice how quickly the functions become very small as they repeatedly lose value because of the fraction compounding on them.*1583

*1/10 becomes very small by the time it has gotten to just 2; we have 1/10 ^{2}, which is equal to 1/100.*1590

*So, it becomes very, very small: by the time we are at (1/10) ^{10}, we are absolutely tiny.*1598

*Once again, it looks like it touches the x-axis, but that is just because it is a picture.*1604

*It never actually quite gets there; there is always a thin sliver of numbers between it.*1608

*But it gets very, very, very close; they will all become very, very small as the fraction on fraction on fraction compounds over and over.*1611

*Bits get eaten away each time the fraction hits, so it gets smaller and smaller and smaller.*1621

*But notice: if we go the other direction, we end up getting very large, just like normal exponential functions that grow, where a was greater than 1.*1626

*They got small when they went negative; they grew when they went positive,*1635

*because when they went negative, they flipped; we have that same idea of flipping.*1639

*If we have 1/10 to the -2, well, that is going to be 10/1 squared, which is equal to 100.*1642

*And that is why we see it blow up so quickly--it becomes very, very large, because we go negative for decay things.*1651

*But we will normally be looking at it as we go forward in time, which is why we talk about decay,*1657

*and things that are greater than 1 being growth, because we are normally looking at it*1661

*as we go forward--as we go to the right on our horizontal axis.*1664

*All right, let's look at some examples.*1668

*A bank account is opened with a principal of $5000; the account has an interest rate of 4.5%, compounded semiannually (which is twice a year).*1670

*How much money is in the account after 20 years?*1678

*So, what do we need? We go back and figure out the function we are using.*1680

*The formula is the one for interest compounded; so it is our principal, times 1 plus the rate, but divided by the number of times it occurs,*1685

*and then also raised to the number of times it occurs in the year, times the number of years that pass.*1696

*So, what are the numbers we are dealing with here?*1701

*We have a principal of $5000; we have a percentage rate of 4.5%, but we need that in decimal, so we have 0.045.*1703

*And what is the amount of time? The amount of time is 20 years.*1717

*If we do this with it going semiannually, twice a year, when we look at that, it will be n = 2.*1722

*a at 20 =...what is our principal? $5000, times 1 + the rate, 0.045 divided by the number of times it occurs in the year;*1730

*it occurred twice; n = 2, so divide by 2; raise it to the 2, times how many years? 20 years.*1743

*We go through that with a calculator; it comes out to 12175 dollars and 94 cents.*1751

*Now, what if we wanted to compound more often--what if it had been compounded quarterly or monthly or daily or continuously?*1763

*If it was compounded quarterly, it would occur four times in the year--every quarter of the year, every season--so n = 4.*1770

*So, we have 5000 times 1 + 0.045/4; that will be 4 times 20; we use a calculator to figure this out.*1779

*It comes out to 12236 dollars and 37 cents.*1792

*So, notice that we end up making a reasonable amount more than we did when it was compounded just twice in the year.*1800

*We are making about 50 dollars more--a little bit more than 50 dollars.*1805

*What if we have it do it monthly? How many months are there in a year?*1810

*There are 12 months in a year, so that would be n = 12.*1813

*5000 is our initial principal, times 1 + our rate, over 12 (I am losing room)...12 to the t...12 times t; so what is our t?*1817

*Our t was 20; sorry about that...12 times 20.*1829

*That will come out to be 12277 dollars and 33 cents.*1836

*What if we have it at daily--how many days are there in the year?*1845

*There are 365 days in a year, so that will be an n of 365.*1849

*So, at 365, we have 5000 times 1 + 0.045/365 (the number of times it occurs--365--the number of times it occurs in the year);*1855

*we had 20 years total; we simplify that out; we get 12297 dollars and 33 cents.*1871

*And what if we managed to do it every single instant--we actually had it compounding continuously?*1881

*Well, if n is equal to infinity, we are no longer using this formula here.*1886

*We change away from this formula, and we switch to the Pe ^{rt} formula, because that is what we do for compounded continuously.*1890

*That is going to be 5000 times e; what is our rate? 0.045; how many years? 20 years.*1898

*Once again, we punch that into a calculator: there will be an e key on the calculator--*1906

*you don't have to worry about memorizing that number that we saw earlier, because there is always an e key.*1909

*5000 times...oops, let's just get a number here; we are not going to end up doing this number, because it would be hard to do.*1915

*We will use a calculator; so let's just hop right to our answer.*1926

*We get 12298 dollars and 2 cents.*1928

*Finally, I would like to point out: notice that we ended up seeing reasonable amounts of growth*1935

*when we jumped from going only semiannually (twice a year) to four times a year.*1940

*And we also saw an appreciable amount of increase when we went from four times a year to twelve times a year-- when we went to monthly.*1945

*We got a jump of a little over 40 dollars.*1953

*When we managed to make it up to daily, we got a jump of about 20 dollars.*1956

*But going from daily to every single instant forever only got us a dollar.*1960

*So, we get better returns the more often it happens; but they end up eventually coming to an asymptote.*1964

*It increases asymptotically to this horizontal...it eventually stabilizes at a single value.*1970

*So, you won't see much difference between an account that compounds every single day and an account that compounds every single instant.*1976

*There won't be a whole lot of difference.*1984

*It is much better to have daily versus yearly, but daily versus continuously is not really that noticeable.*1986

*The second example: The day a child is born, a trust fund is opened.*1992

*The fund has an interest rate of 6% and is compounded continuously.*1996

*It is opened with a principal of $14000; what is the fund worth on the child's eighteenth birthday?*1999

*What formula will we be using? We will be using Pe ^{rt}.*2004

*The amount that we have in the end is equal to the principal that we started with, times e to the rate that we are at times t.*2007

*What is our principal? Our principal is 14000 dollars. What is our rate? Our rate was 6%.*2014

*We can't just use it as a 6; we have to change it to a decimal form, because 6 percent says to divide by 100; so we get 0.06.*2023

*Finally, what is the amount of time that we have?*2030

*In our first one, we are looking at a time of the eighteenth birthday--so 18 years; t = 18.*2032

*A principal of $14000 times e to our rate, 0.06, times the amount of years, 18 years--*2039

*we plug that into a calculator, and we see that, on his eighteenth birthday, the child has managed to get 41225 dollars and 51 cents.*2050

*So, that is pretty good; but what if the child managed to not need the money--didn't really want the money--*2062

*wanted to save it and maybe use it to buy a house when he was 30 (or put down a good down payment on a house when he was 30)?*2067

*At that point, if he was 30 before he took out the money, the child would have 14000;*2073

*it is the same setup, but we are going to have a different number of years--times 30.*2079

*That would end up coming out to 84000; it has more than doubled since he was 18--pretty good.*2083

*So, it has more than doubled; he has managed to make $84000 there.*2092

*That is not bad--he could get a good down payment on a house with that, so it is pretty useful.*2097

*But if he really didn't need the money--if he managed to not spend that money,*2102

*and he said, "I will use it as a retirement fund; that way I won't have to invest for my retirement at all--I already have it set up."*2105

*How much would he end up having at the age of 65?*2110

*We have 14000--the same setup as before--times e to our rate, 0.06, times our new number of years we are doing--it is 65 years.*2114

*And you would manage to have a huge 691634 dollars and 29 cents.*2124

*So, this points out just how powerful compound interest was.*2134

*We managed to start at 14000 dollars; but if we can avoid touching that money,*2137

*if we can just leave it for a very long time, we can get to very large values as the interest compounds on itself over and over again.*2142

*In 65 years, which is a very long time, we managed to grow from 14000 dollars to 691634 dollars--a lot of money.*2148

*And this gives us an appreciation for how important it is to make investments for retirement at an early age.*2162

*It is difficult when you are young; but if you manage to invest when you are young--*2168

*if you can wait on spending that money now--it can grow to very large amounts by the time you want to spend it to retire.*2172

*So, that is the benefit of investing early--being able to do that.*2177

*Also, it shows just how great, how useful, an interest rate is.*2181

*If that 6% was bumped up to 8% or 10%, we would see massive increases.*2185

*You can get a lot of increase if you can just get that percentage rate up another point or two--it is pretty impressive.*2189

*All right, the third example: The population of yeast cells doubles every 14 hours.*2194

*If the population starts with 100 cells, how many cells will there be left in two weeks?*2199

*So, this isn't compound interest, and it isn't continual growth, like we had before.*2205

*We might want to build our own here.*2210

*The population is doubling, so let's say n is the number of cells after some time.*2212

*We will set it up as a function--that makes sense; we are in "Exponential Function Land" right now.*2217

*So, n(t) is equal to...well, how many cells did we start with?*2221

*We started out with 100 cells, and we were told that it doubles.*2226

*So, we are going to have some "times 2," because we multiply it by 2.*2231

*How often does it do that? It does it every 14 hours.*2234

*So, if we have our number of hours, t = number of hours, t divided by 14 will be how many times it has managed to double.*2237

*After 14 hours, we have multiplied by 2 once.*2249

*After 28 hours, we have multiplied by 2 twice; we have 2 times 2 at 28 hours.*2252

*So, let's do a quick check and make sure that this is working out so far.*2258

*So, if we had n at 14 hours, we would have 100 times 2 ^{14/14}, which would simplify to 100 times 2^{1}.*2262

*So, we would get 200, so that part checks out.*2273

*Let's try one more, just to be sure: n(28)...if we had double double, then we know that we should be at 400, so we can see what is coming there.*2276

*So, 2 times 28/14...that simplifies to 100 times 2 ^{2}, which is equal to 100 times 4, or 400.*2285

*So, that checks out, as well; it passes muster--this makes sense as a way of looking at things.*2295

*So, as long as we have the amount of time we spent and the number of hours,*2300

*then we can see how many cells we have after that number of hours.*2304

*Now, we were told to figure out how many there will be in 2 weeks.*2308

*And we can assume that none of the cells die off, so the number just keeps increasing.*2313

*It is a question of how many times the population has gotten to double.*2316

*If that is the case, what number are we plugging in--it is n of how many hours?*2320

*Is it 2? No, no, it is not 2! Well, how many weeks...oh, 14 days? No, it is not 14.*2324

*What were we setting this up in? t was set in number of hours.*2330

*So, the question is how many hours we have on hand.*2334

*Let's first see how many hours 2 weeks is: how many days is that?*2337

*Well, that is going to be 2 times...how many days in a week?...7, so that is 2 times 7 days.*2344

*How many hours is that? 2 times 7 times 24, or 14 times 24 hours, which we could then figure out with a calculator, and get a number of hours.*2349

*But we can actually just leave it like that, which (we will see in just a few moments) is a useful thing to do,*2361

*because we notice that there is a divide by 14 coming up; maybe it would be useful to just leave it as 14 times 24--a little less work for us.*2365

*So, 14 times 24...now notice: 14 times 24 is the number of hours in 2 weeks.*2373

*That is why we are plugging that in, because once again, the function we built,*2384

*our n(t) function that we built, was based on hours going into it.*2388

*We can't use any other time format.*2392

*100 times 2 to the 14 times 24 (is the number for t), divided by 14; look at that--the 14's cancel out.*2395

*We can be a little bit lazier--that is nice.*2406

*100 times 2 to the 24: we plug that into a calculator, and we get a huge 1677721600 cells.*2408

*That is more than one and a half billion cells: ones, thousands, millions, billions.*2425

*So we are at 1.6 billion cells--actually, closer to 1.7 billion cells.*2432

*This gives us a sense of just how fast small populations are able to grow.*2440

*And that is how populations grow: they grow exponentially, because each cell splits in half.*2443

*So, if we have one cell split in half to 2, and then each of those splits in half to 4,*2449

*and each of those splits in half to 8, this is going to do this process of exponentiation.*2453

*We are doing this through doubling, so we are going to see very, very fast growth.*2458

*And we actually see this in the real world.*2461

*We could also write this, for ease, as...we have 1, 2, 3, 4, 5, 6, 7, 8, 9...so that is the same thing...*2464

*we could write it as approximately 1.67x10 ^{9} cells,*2474

*so that we can encapsulate that information without having to write all of those digits.*2482

*That is scientific notation for us; all right.*2486

*The fourth and final example: The radioactive isotope uranium-237 has a half-life of 6.75 days.*2488

*Now, what is half-life? We would have to go figure that out, but luckily, they gave it to us right here.*2495

*Half-life is the time that it takes for one-half of the material of our isotope to decay and break down--to go through a process of decay.*2499

*If you start with one kilogram of U-237, how much will have not decayed after a year?*2508

*So, we are saying that, after 6.75 days, we will have half of a kilogram.*2515

*We start with one kilogram, and we know that, after every 6.75 days, we will have lost half of our starting material.*2522

*So, we will go down from one kilogram to half of a kilogram that has not decayed.*2529

*So, let's see if we can figure out a way to turn this into another function.*2533

*The...let's make it amount...the amount of our isotope that has not decayed, based on time,*2537

*is equal to...how much did we start with? We started with 1 kilogram; times...what happens every cycle?*2545

*1/2...we halve it every time we put it through a cycle; so how fast is a cycle?*2551

*The number of days--we will make t into the number of days, because we see that we are dealing with days, based on this here.*2559

*So, t divided by 6.75--let's do a really quick check.*2567

*We check, because we know that, after 6.75 days, we should have 1/2 of a kilogram.*2572

*So, let's check that by plugging it in: a(6.75) is going to be 1 times 1/2 raised to the 6.75 over 6.75,*2580

*which is the same thing as just 1/2 to the 1, which equals 1/2.*2592

*So sure enough, it checks out--it seems like the way we have set this up passes muster,*2596

*because it is going to divide by half every time the 6.75 days pass.*2601

*So, if we plugged in double 6.75, it would divide by half twice, because it would be 1/2 squared.*2604

*It seems to make sense; we have set it up well; and we can see that this also can be just written as 1/2 times...*2610

*let's just leave it as it is; it gives us a better idea of how this works in general, for half-life breakdowns.*2616

*So, now we are going to ask ourselves how long--what is the time that we are dealing with?*2621

*In our case, t is one year; what is one year in days (because we set up our units as days,*2625

*because that is what our half-life was given to us in)?*2632

*One year is 365 days; so at the end of that, we plug in 365 = 1 (the amount that we started with),*2634

*times...the half-life will occur every 6.75 days (and we are still having 365 days go in).*2643

*We plug that all into a calculator, and we get the amazingly tiny number of 5.273x10 ^{-17} kilograms--a really, really, really small number.*2652

*To appreciate how small that is, let's try to expand it a bit more.*2669

*1 kilogram is 1000 grams; so that means that a kilogram is 10 ^{3} grams.*2672

*We could also write this as 5.273x10...if it is 1000 grams for a kilogram, then that means we are going to increase by 3 in our scientific exponent.*2681

*So, in the scientific notation, we are now at 5.237x10 ^{-14} grams of our material.*2695

*which, if we wanted to write this whole thing out...we would be able to write it as 0.00000 (five so far) 00000 (10 so far) 000 (13)...*2701

*and let's see why that is the case--we can stop writing there--because if we were to bring that 10 ^{-14} here,*2720

*(and remember, it is in grams, because we had grams here), that would count as 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14,*2728

*because we can move the decimal places 14 times to the right by having 10 ^{-14}.*2740

*And that is how that scientific notation there is working.*2745

*Or alternatively, we could also write this with kilograms as the incredibly tiny 0.00000000000000005273 kilograms.*2749

*And if we counted that one out as well, we would have 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17...*2767

*so we have that 5.273x10 ^{-17} kilograms there, as well.*2779

*So, it is much easier to write it with scientific notation; that is also probably what a calculator would put out,*2783

*because it is hard to write a number like this, this long, on a calculator.*2788

*So, we are much more likely to see it in scientific notation, 5.273x10 ^{-17} kilograms,*2792

*which is an absolutely miniscule amount of radioactive material left, considering that we started at 1 kilogram.*2799

*That shows us how decay works.*2805

*All right, cool: we have a pretty good base in exponential functions.*2807

*Next, we will see logarithms, and see how logarithms allow us to flip this idea of exponentiation.*2810

*And then, in a little while, we will see how logarithms and exponential functions...how we can oppose the two against each other.*2814

*It is pretty cool--we can find out a lot of stuff with this.*2819

*All right, we will see you at Educator.com later--goodbye!*2821

1 answer

Last reply by: Professor Selhorst-Jones

Fri Mar 25, 2016 5:12 PM

Post by Jay Lee on March 21, 2016

Hi,

How would you simplify (x^(5/4)y^(-5/4))/(x^(-1/2)y^(7/4)x^(2)y^(2))^(5/3)? The question requires the answer to contain only positive exponents with no fractional exponents in the denominator. I tried solving the question and ended up with a fraction in the denominator, so I changed those to the "root" form, but the teacher wouldn't accept that answer either. I solved until: (y^(1/2)x^(3/4))/(y^(8)x^(2)), but I don't know how to simplify it further.

Thank you so much:)