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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 the base. [The base is the thing being raised to some exponent.]f(x) = a^{x},  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 accountthe 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.
 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
 The symbol e represents a number in the same way that π does. Also, just like π, e is irrational: its decimal expansion never ends and never repeats.
e = 2. 718 281 828…  Clearly, we can't use all the digits of e because there are infinitely many of them. However, the more of them we use, the more accurate our calculation will be. Since we need three decimal places of accuracy, let's begin by using at least six decimals from e: 2. 718 281. [This won't be perfectly accurate, but it will be accurate enough to find e^{3} to three decimal places. If we want more accuracy, we can use more decimals from e.]
 Plug the approximation into a calculator:
Thus, we round to three places, saying e^{3}≈ 20.086. [Notice that the above approximation is not perfect. As we use more decimals, we get a more accurate result. Using a massive number of digits, we could find thate^{3} ≈ 2.718281^{3} = 20.085 518 558 …
See how the highly accurate value for e^{3} above starts to give different values after the fourth digit. While we can get good accuracy with a modest approximation of e, if we really need a massive amount of accuracy, we have to start with a lot of accuracy in the value we use for e.]e^{3} = 20.085 536 923 …

 To fill in the table, plug in each value to the function.
 For 3^{−1}, remember that raising anything to the negative exponent causes it to "flip" to its reciprocal.
 For 3^{0}, remember that raising anything to the 0 causes it to become 1.
 As you fill out the table, notice how very quickly the function grows. Exponential functions become massively huge very, very quickly.


 An exponential function is a function of the form f(x) = a^{x}, where a is some base. Thus, to answer the question, we need to figure out what a is.
 The fact that f(0)=1 is not much help to us. As long as the function is strictly in the form a^{x}, then any value of a will still produce 1. This is because raising any number to the 0 causes it to become 1. So we can't get much useful information from the first entry in the table.
 The second and third entries are much more useful, though. We know f(2) = a^{2} = 22.09 and f(3) = a^{3} = 103.823. Notice that a^{2} ·a = a^{3}. Thus,
a^{2} ·a = a^{3} ⇒ 22.09 ·a = 103.823 ⇒ a = 4.7  Finally, now that we know a=4.7, we can doublecheck our answer. Currently, we have f(x) = 4.7^{x}. From the table, we know f(4) should be 487.9681. Check using a calculator that 4.7^{4} gives the same value. Indeed, it does, so our answer is correct.
 Like graphing any function, we can always use brute force: calculate and plot a bunch of points, then draw a graph from those points. Approaching it that way, we might create a table such as the below:
x f(x) −5 −16.9999 −4 −16.9998 −3 −16.9990 −2 −16.996 −1 −16.984 0 −16.938 1 −16.75 2 −16 3 −13 4 −1 5 47  Alternatively, we could examine the structure of the function. Notice that f(x) = 4^{x−2}−17 is similar to the expression 4^{x}. From looking at graphs of other exponential functions, it might be very easy for us to graph 4^{x}: it would be 1 at x=0, 4 at x=1, and shoot up rapidly as we go farther to the right. If we examine 4^{x} as it goes to the left, it would approach the xaxis asymptotically. Now notice that 4^{x−2} would have the same graph as 4^{x}, except it would be shifted 2 units to the right. Next, 4^{x−2}−17 would be shifted a further 17 units down. [If you're not sure how to see this, check out the lesson on function transformations.] Thus, f(x) = 4^{x−2} −17 will have the same shape as 4^{x}, but shifted 2 units right and 17 units down.
 Either way that we approach understanding the shape of f(x), we now need to graph it. Notice that f(x) moves very quickly vertically for a small amount horizontally. Thus, we might want to choose graphing axes that are very tall for a small width.
 Because the base of the exponent is smaller than 1, the value of the function will become very small for positive x. As the fractions multiply each other more and more, the result will be tiny. On the other hand, when we have a negative x, the negative exponent will cause the fraction to flip, and thus allow the function to grow.
 To help us see how the function graphs, calculate various points:
x g(x) −6 11.39 −4 5.06 −2 2.25 0 1 2 0.44 4 0.20 6 0.09  Plot points and connect with curves. Notice that as x moves very far to the right, the function will asymptotically approach a value of 0. On the other hand, as x moves to the left, the function will grow very rapidly.
 We can describe the amount of money, A, in such an account with an exponential function:
where P is the principal in the account, r is the interest rate (as a decimal), n is the number of times the account compounds per year, and t is the number of years elapsed.A(t) = P ⎛
⎝1 + r n⎞
⎠nt
,  For this problem, we have
The only thing left is how many times the account compounds per year, and the question asks us to figure that out for three different compounding schedules.P=10 000, r = 0.047, t = 25.  Those compounding schedules are as follows: annuallyonce a year ⇒ n=1; semiannuallytwice a year ⇒ n=2; dailyevery day, so 365 times in a year ⇒ n=365. Use this along with the information above to figure out how much money is in the account for each situation.
Annually: 10000 ⎛
⎝1 + 0.047 1⎞
⎠1·25
= 10000 (1.047)^{25} = 31 525.87 Semiannually: 10000 ⎛
⎝1 + 0.047 2⎞
⎠2·25
= 10000 (1.0235)^{50} = 31 944.22 Daily: 10000 ⎛
⎝1 + 0.047 365⎞
⎠365·25
= 10000 (1.000128767…)^{9125} = 32 378.98
 If something is being compounded continuously, the amount (A) of money after t years is
where P is the principal, r is the interest rate (in decimal), and e is the natural base.A(t) = Pe^{rt},  The principal investment is P=25 000 and the rate is r=0.07. If she puts the money into the investment at age 45, then retires at age 65, that means the investment has t=20 years to mature. Thus, its value is
25000 e^{0.07 ·20} = 25000 e^{1.4} = 101 380.00  For the other ages, we just need to figure out new values for t. If she invests at age 35, then that means t=30 years until retirement:
If she makes the investment at 25, then she has t=40 years:25000 e^{0.07 ·30} = 25000 e^{2.1} = 204 154.25
[Notice the massive increase in the value of the investment. Because compounding investments are based on exponential functions, the earlier the investment can be made, the greater its final value will wind up being.]25000 e^{0.07 ·40} = 25000 e^{2.8} = 411 116.17

 We have a formula to determine the value of the car after t years of use. To find the value of the car after one year of use, simply plug in t=1:
p(1) = 26650 (0.87)^{1} = 23 185.50  To find his net cost upon selling the car at seven years, we first need to know how much he sells the car for. Figure out the value of the car after seven (t=7) years:
p(7) = 26650 (0.87)^{7} = 10 053.84  Now that we know Bryan is able to sell the car for $10 053.84, we can calculate his net cost in owning the car. While it cost him $26 650 to purchase the car in the first place, he ultimately managed to sell it for some money, thus lowering the net cost of owning the car. To find the net cost, simply subtract the selling price from the original cost:
26650−10053.84 = 16 596.16
 Notice that we are told the population doubles after a certain time interval. This is equivalent to multiplying the starting population by 2 after a time interval passes. After the next time interval, it will multiply by 2 again, and so on. Thus, if we say the number of time intervals is n, and the starting population is P, the amount A of bacteria will be
A = P·2^{n}.  Now we need to figure out how many time intervals have elapsed. We know that one interval occurs every 26 minutes. The total time is 12 hours, which is 720 minutes. Thus, the number of intervals that have occurred is
n = 720 26.  We can plug that in to the above to figure out the amount of bacteria after 12 hours:
A = 50 ·2^{[720/26]} = 10 843 894 130
 Notice that we are told the quantity halves after a certain time interval. This is equivalent to multiplying the starting amount by [1/2] after a time interval passes. After the next time interval passes, it will multiply by [1/2] again, and so on. Thus, if we say the number of time intervals is n, and the starting quantity is P, the amount A of quantity in the end will be
A = P· ⎛
⎝1 2⎞
⎠n
.  Now we need to figure out how many time intervals have elapsed. We know that one interval occurs every 5 730 years. The total time is 30 000 years. Thus, the number of intervals that have occurred is
n = 30 000 5 730.  We can plug that in to the above to figure out the quantity of not decayed C14 after the passage of 30 000 years:
A = 2.3 · ⎛
⎝1 2⎞
⎠[30000/5730]
= 0.061
*These practice questions are only helpful when you work on them offline on a piece of paper and then use the solution steps function to check your answer.
Answer
Exponential Functions
Lecture Slides are screencaptured 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
Precalculus with Limits Online Course
Transcription: Exponential Functions
Hiwelcome 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 number0005
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 exponentiatedwhatever is going through this process of having an exponentthat 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 thatwe can't divide by 0.0104
So, that is not allowed, so that means a = 0once 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 xvalues, 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 numberswe 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 iswhatever the problem asks forby 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 thingsomething random.0223
They might have a carat, which says...if I have 3^{6} (not with an aI 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 stepbystep 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 calculatornot 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 fromthere is some pretty interesting stuff in calculus.0354
All right, now, if we can evaluate at any placeif we can compute what these values of exponential functions are0361
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 0they 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 xaxis; it isn't quitethere 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 faroff 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 2it 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 twentyfirst 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 twentyeighth 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 worriedhe 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 fortyfirst 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 goingif we let it run all the way to the sixtyfourth 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 greedydon'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 takeaway 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 investment0922
(the amount that you put in the bank), the following year you would have that $100 still0931
(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 10your 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 moneythe 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 accountthe 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 times1112
that the interest has hit, over and over and overour principal times the 11119
(because the bank lets you keep what you started with), plus the interest in decimal form,1125
all raised to the tthe 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 interest1136
in a lump sum at the end of the yearthey 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 twiceinayear 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 possibleevery minuteevery secondevery instant1271
if we had it happening continuouslyabsolutely 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 infinityas this becomes larger and larger1288
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 infinitywhat 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 growing1423
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 instantthat 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 amounthowever 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 herethis "Pert" thingthis can be used for a wide variety of things1493
that grow or decay continuouslythings 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 importantthis 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 things1516
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 forward1550
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 < 11562
if a is between 0 and 1it is a fractionit 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 xaxis, 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 quicklyit 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 it1661
as we go forwardas 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 oftenwhat 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 yearevery quarter of the year, every seasonso 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 morea 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 dailyhow 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 occurs365the 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 instantwe 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 calculator1906
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 growth1935
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 birthdayso 18 years; t = 18.2032
A principal of $14000 times e to our rate, 0.06, times the amount of years, 18 years2039
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 moneydidn't really want the money2062
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 yearstimes 30.2079
That would end up coming out to 84000; it has more than doubled since he was 18pretty good.2083
So, it has more than doubled; he has managed to make $84000 there.2092
That is not badhe 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 moneyif 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 allI already have it set up."2105
How much would he end up having at the age of 65?2110
We have 14000the same setup as beforetimes e to our rate, 0.06, times our new number of years we are doingit 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 dollarsa 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 young2168
if you can wait on spending that money nowit can grow to very large amounts by the time you want to spend it to retire.2172
So, that is the benefit of investing earlybeing 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 twoit 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 functionthat 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 musterthis 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 init 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 24a 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 thatthe 14's cancel out.2395
We can be a little bit lazierthat 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 cellsactually, 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 uranium237 has a halflife of 6.75 days.2488
Now, what is halflife? We would have to go figure that out, but luckily, they gave it to us right here.2495
Halflife is the time that it takes for onehalf of the material of our isotope to decay and break downto go through a process of decay.2499
If you start with one kilogram of U237, 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 dayswe 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 6.75let'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 outit 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 halflife breakdowns.2616
So, now we are going to ask ourselves how longwhat 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 halflife 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 halflife 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} kilogramsa 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 casewe can stop writing therebecause 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 coolwe can find out a lot of stuff with this.2819
All right, we will see you at Educator.com latergoodbye!2821
1 answer
Last reply by: Professor SelhorstJones
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:)