welcome to another video today we're going to talk about exponentials i know that you've seen them before you've probably heard them before you might even graph them but we're going to look at them under kind of the why they look the way that they do and how we're going to use them for logarithms in just a little bit so we really want to get a good feel of why these are created the way they are why they look the way they do why they are one-to-one functions and that they will create an inverse graph that we call a logarithm in just a little bit so deeper than just a cursory understanding of hey this is what exponential looks like so we'll talk about right now what an exponential is but that should be the back of our thoughts as why they look the way that they do i'm going to create two graphs at the same time we'll talk about key points domain talk about range horizontal asymptote and then we'll move on from that so what an exponential is an exponential is different from a polynomial so a polynomial what it does it takes a variable and raises it to a constant exponent has a lot of different terms you can do that and adds them all up an exponential switches that it takes a constant that we call a base that's always positive and not one not zero and it takes it to an exponent which is a variable which is a very interesting concept we get a lot of really good population modeling from exponential type of growth so that's what this means so when we see a function where we have a base that's a constant raised to a power or an exponent this is a variable that's an exponential function a is always positive it's not zero and it's not one for the following reasons if you had a having the ability to be a negative what would happen what would happen is we'd have this negative number raised to the first power would give you a negative raised to the second power would give you a positive and we get this very disjointed dis non-continuous uh graph we have a lot of continuities in that it'd be very difficult to even graph really consider what's going on and so we disallow that instead we say hey our base is going to be positive it's not going to be 1 why why can't it be 1 well if you take 1 to any power you're going to get one it's going to create this horizontal line at y equals one which would be great if that's all we were doing said oh yeah every differential is this but that's just that's not true what we're going to understand about this is that as our one sorry as our as our base gets bigger than one we're going to get an increasing graph and as our base gets less than one but not negative remember it has to be between zero and one basically we're going to get this decreasing graph and we will explore that as we continue so that is what an exponential comes from is this base that's a constant that is positive not one raised to this variable that is that is your exponent and that's going to be changing so now we're going to move on to talk about graphs you see there's two cases when we talk about exponentials even though our a is positive always zero we have this one this value of one that causes our graphs to either always increase or always decrease depending on whether our base is above or below that why why is that the case and this is going to be the fundamental reason why our graphs look different here it is if you take one to any power true enough you get out one if you take numbers that are greater than 1 and you take them to positive exponents those numbers are going to grow so if i take numbers like let's use this for instance let's use 2 to the x and i take two to the first i'm gonna go up two two to the second i'm gonna get a four two to the third i'm gonna get eight and then two the four is sixteen if i take numbers that are more than one and i take them to positive exponents these are going to grow correct numbers greater than one raised to powers that are positive are going to grow numbers that are less than one so between zero and one like fractions like one one-half and it does have to be in parentheses it's the whole fraction being raised to that power if i take a base of one-half or numbers between zero one and i start taking them to positive exponents what happens well one half to the first power is one half sure but one half to the second is one fourth one after the third is one eighth when that fourth is 1 16 they get smaller this is the reason why our graphs do two different things is because values that are greater than one when i take them to positive exponents grow and values less than one when i take in positive exponents shrink they decrease how about negative numbers that are greater than one when i take them to negative exponents create reciprocals for us so that that was a concept a long time ago negative exponents don't create negative values they create fractions or reciprocals of fractions if we start with fractions so two to the negative one doesn't create negative two it creates one half two to the negative two doesn't create uh negative four it creates one-fourth and so this is going to climb and likewise here one-half to the negative one doesn't create negative one-half rates two this is going to fall which is the reason why we have these two different graphs we're going to explore that right now in a little bit more detail we're going to come up with the fact that every single exponential has some key points and a horizontal asymptote and why that is so i talked about some values we're now going to plot them we're going to see why this graph climbs where you see why this graph falls we'll get our key points from that so let's plug in some values if we start by plugging in let's say 0 0 is always a great value to plug in because it gives us our y-intercept if i plug in 0 2 to the 0 power is going to give me 1. how about when i plug in 1 2 to the first power is going to give me 2. so i'm getting out 0 1 1 2 if i evaluated for two i'd get out four now we can use a lot more values if we wanna plug in three two to the third would give us eight four to the fourth would give us 16. five two to the fifth would give us 32. this is going to be a pretty steeply climbing graph now if we about if i evaluate some negative values like negative one two to the negative one power two to the negative one equals one over two that's gonna be one half and two to the negative two would give us 1 over 2 squared well that's 1 4. and so we don't climb back like a parabola we don't get negative we get these values positive one-half and positive one-fourth now here's a big question if i plug in negative three in your head you should be thinking see two negative third that's one over 2 cubed that's 1 8. and then if we're plugging negative 4 of that b that'd be 1 16 then 1 32nd then 1 64. is this ever going to hit zero is it even possible is it even possible to take a value and put it here and make this zero is two to any power equal to zero well you try zero right but two the zero power is giving you one not zero it's impossible for us to plug in a number here that's going to cause this to be zero in fact this is big it's impossible for us to plug a value in here that even makes this negative i tried positives and gave me really big values i tried zero it gave me one i tried negative it gave me fractions four bases that are greater than one it's just giving me fractions but they're positive this has no ability to give me an output that's negative now that's interesting that says that we can plug in any number right our domain is going to be all real numbers but i can't get out every number i can't even get out zero oh i can't even get out of zero i can't even get out negatives and therefore since i can't get out negatives and i can't get out zero we're going to have a horizontal asymptote at y equals 0. this graph is always increasing that is what an exponential looks like if your a is more than one we're going to remove the two and talk about in general when a is greater than one in just a minute before we do that we're going to answer this this graph so let's take a look at one half to the x let's start plugging in some values exactly like we did if we evaluate zero and i take one half to the zero power so i'm evaluating x equals zero one half of zero anything to the zero powers if you have a base that is a a positive number is going to give you out one well that's the same exact output is here now it says ask another question is it always true that if i take some sort of constant and i evaluate it for x equals zero am i always going to get out one the answer is yes every single one of these exponentials before you start shifting it manipulate it do transformations every one of them if you evaluate x equals zero it will give you out one why because any constant raised to the zero power is one that's huge that's going to be a key point for us now let's move on what happens if i plug in one evaluate for one if i evaluate for one one half to the first power remember that's our variable one half of the first power is going to give me out one half anything raised to the first power gives you back what you plugged in so i get this point one comma one half way over here i evaluated one and i get out two here i evaluated one i got a one half is it true that every time you evaluate an exponential for the value x equals 1 you'll get back the base is that always true answer is yes that's always true if i plug in 1 here i'm going to get the base 2. if i plug in 1 here i'm going to get the base one-half if i plug in 1 here i'm going to get the base a those are the two and only two key points you ever need for exponentials in order to graph them especially with transformations you need the key points 0 1 and 1 comma the base don't go any further until you understand that those two key points are going to show up on every exponential before we shift it you need that you need to understand that with an exponential if i evaluate for zero a to the zero power gives me one that's big and if i evaluate for one a to the first power gives me back whatever my base is these two key points are how we're going to graph in the next video when we talk about transformations 0 1 and 1 comma whatever your base is let's continue if i plug in 2 well 1 half to the second power is going to give me 1 4. same question is this one can we ever get this to touch zero touch the x-axis or be negative well if i plug in three i'm gonna get one-eighth forming at one sixteenth five and one thirty seconds six and so forth and so on i can't ever make this negative in fact if you take a positive number to any exponent you're not going to get a negative we explored that here this can't be negative it is going to get closer and closer to zero as an output well wait a second that's this idea of a horizontal asymptote again so we have the same idea that both of these graphs and every exponential in general is going to have a horizontal asymptote on the x-axis before you start shifting it on the x-axis or at y 0. now what happens when we plug in negatives if we evaluate for negative one one-half to the negative one power there's a nice property of exponents it says if you evaluate a fraction to a negative exponent you can reciprocate the fraction and it will change the sign of your exponent so one half to the negative one is equivalent to two over one to the first power that means we're going to get two and likewise if that's a negative two i would reciprocate and have two over one squared will that give you 4. these functions are in fact symmetrical about that y-axis we can see that this one is increasing all the time this one's decreasing all the time you can see they both have a key point of zero one and one comma whatever the base is here's your zero one here's your one comma your base here's your zero one and here's your one comma whatever your base is this is true in general for exponentials until you start shifting them around they both have a horizontal asymptote at y equals zero that's also true for exponential so this is kind of a big deal understanding where these things came from why they look the way they do and especially identifying what they're going to look like for your base so if your base is more than one they're going to increase if your base is less than one they're going to decrease um in general so in general this is if you're uh if this a was just greater than one not specifically equal to two and we started erasing some of this stuff and saying well i don't know exactly what my base is here's how this graph is going to look if your a is more than one you're still going to get zero one but you'll just have some sort of a key point that's that's above 0 1 y well if a is more than 1 and you plug in and you plug in let's say 1 here you're going to get 1 comma a if your a is above 1 it will be one comma something above one it would have to be above that value if a is above one if a is below one so we'll change this to not necessarily one half if your a is less than one then when you plug in one you're going to get something one comma something less than one that's below here that's why that has to drop if both of these these functions whether a is above one or below one if both these functions have that point zero one then the a being greater than one would climb and a b and less than one would have to fall in general that's true so here's the way i like to think of it for for exponentials if a were to equal one which is not going to if a were to equal one you would get a horizontal line one to any power is going to give you the horizontal line that's a constant if your a is greater than 1 that raises the positive side of your y-axis and lowers the negative side if a is less than 1 it's reverse so a more than one yeah start climbing and the more you're more than one the steeper it's going to climb a less than one you're gonna start falling the more you're less than one you're going to start really falling dramatically that's that's a cool thing one other thing to to see here is the um yeah i don't know now i'm not going to show it to you but uh there's some even evenness going on with the switch of a sign we can get that a reflection of these graphs which is kind of neat to see this is key this is what we want to know is what we want to talk about in general that's true for your your key points that's true for your horizontal asymptote which we'll explore in the next section other than that our domain for our exponentials you should see this you can plug in any number we plugged in zero we plug in positives we plugged in negatives there's nothing wrong with this your range is not all real numbers your domain you can plug in anything all real numbers your range is strictly positive you cannot get up zero and you cannot get out negative numbers from an exponential before you shift it so your domain is from zero to positive infinity but it's not included in zero that's another way to say all real numbers range we get out things very close to zero which is why we don't use a bracket all the way to positive infinity in both of these cases all right so the last thing and a very important thing we have a special number in math we have lots of special numbers in math one of them is pi one of them is e now i know you're familiar with pi and hopefully you're familiar with e this number e right here that is a constant number it is a value that does not change that's a weird value it's called the euler number euler is spelled with an e and so this is this is based on on him leonard euler it is called the euler number or the natural number and it's about 2.7 now this number we can't write out as 2.7 because it goes forever and it doesn't repeat it's a transcendental number but that is a a very special number and it occurs a lot in nature and in a lot of mathematics so it's really useful to understand an exponential based on that number e so e is approximately 2.7 about two point seven one and it goes forever uh and it doesn't repeat for any length of time that's uh that's modeled by a fraction so it's not a fraction here's the question in which of these two which of these two graphs would best model that would it be a graph that's always increasing or graph that's always decreasing well let's think about that what is the base so is it exponential sure it's got a base what is the base is the base more than one or is the base less than one if the base is more than one it can't look like this one it's going to look like this one in fact it's really close to this this is going to be if we did f of x equals e to the x let's see what these values would do if i plug in 0 am i still going to get 1 yeah always every exponential does that if i evaluate for 0 e to the 0 is still going to give me 1 i'm still going to have that value if i plug in 1 am i still going to get my base out yes except my base is going to be e so e to the first power is going to give me e what that means is that for our graph we get something that if this is one and this is two we'd get about you know i'm gonna change this a little bit evaluate for 0 we get out 1. evaluate for 1 we get out 2.7 of roughly 2.7 this would be one comma e and we can see right there that if every exponential has a .01 and you evaluate for one it gives you out your base then if every exponential has 0 1 if your base is more than 1 you'd get out something above 1. and less than 1 you get something less than 1 when you plugged in 1. so evaluate for 1 if a is bigger than 1 it's up here if a is less than one it's between here that gives you whether it climbs for four bases greater than one or falls for bases less than one it has to do that so this is going to give us a climbing graph it's a little bit steeper than 2 to the x but it's very close because that's 2.7 this is how the graph of f of x equals e to the x looks it still has those key points it still has that horizontal asymptote this is still the domain and range is it going to be useful to try to solve for x-intercepts for an exponential they don't have any um so that that would be kind of a waste of time these are not going to have them until they start shifting down can we can we solve exponentials yes but we need more to it so i told you the idea behind what exponentials are i told you why we're going to need them here's the the key points and key points the key points of what you need to remember for exponentials number one exponentials are caused by taking a base to an exponent that is a variable your base has to be positive if your base is more than one your graph is going to climb and if it's between zero and one your graph is going to fall this is always increasing this is always decreasing they both have the key points zero one because when i take a constant to zero i get out one and they have the the point one comma whatever your bases because if i take the base to the first power i'm gonna get it back they both have a horizontal asymptote at y equals zero that's always true for exponential until i start shifting them finally e to the x is a very special case of an exponential where we have a certain base that's used very often in nature sciences and mathematics and that base is e e is approximately 2.7 and so we are going to get this ever increasing graph this is always increasing on the domain just like that one's always decreasing and finally this is like the the whole punch line of what logarithms are gonna come from is this function one two one use your horizontal line test certainly a function is a one-to-one function yes because no outputs are ever repeated and that's also a one-to-one function because no outputs are ever repeated well if a function is one-to-one it has to have an inverse in a couple videos from now we're going to take that idea we're going to explore it and say yeah this has an inverse in fact we could graph it it would be not hard to graph we'll change our key points from exponentials and get key points of the inverse of an exponential we'll call it a special name called a logarithm and then we'll discover that they solve each other so that's very important because right now there's no way to solve that function if i set this equal to another function or like a constant or something and try to solve it we really can't do it unless some special cases arise so that's what i want you to know right now where they come from and where we're going with i hope you explain it well enough for you to understand that next time we'll practice i know that this while not being vague doesn't let us practice very well how to graph these so we're going to practice next time when we do some transformations of exponential functions hope you're doing well you