 And so it's already time for the third lecture. We're going to look at powers, and those powers are also going to include roots, and then finally add some logarithms. Now we use logarithms as well as powers all the time in data science. So this is a very important lecture. So let's start with powers, such an important aspect of algebra. And let's start with a little example. And if I might say something like three to the power two, what I mean by that is I have three and I multiply it by itself twice. So we have the number that we want to multiply by itself. And in the superscript we put the number that we want to. Now the number of times we want to multiply this with. So for pure definition we'll usually write something like x to the power a. And remember from last time x would usually be used when we refer to any real number. And a would be one of those symbols that we use when we think about whole numbers. And so in this x, since our x would be three, which is a real number, the more important the a is a two, and so that is a whole number. And this is just going to say multiply x by itself eight times. And now we have words that we apply to these different parts. This is called the base. Let's put the base. And this is called the power. Let's put that there. This is the power. It's also known as the exponent in some text books. And so that's what we mean by powers. So we have a little definition there. So let's put a nice little arrow there for us. And we're always going to remember that's the base, that's our power. And we just want to multiply this base by itself. And how many times do we want to do that? Well, we want to do it eight times. And so let's have another little example. Let's do something like, something like four to the power two. So this is going to be raised, multiply four by itself twice. And that's going to be something like 16. It's just a shorthand way of writing all of this. Now that might make sense. You know, if we had to say four to the power 94, you know, writing out four times four times four, 94 times takes a bit of time. But we want to do something a little bit more exciting when it comes to these powers. And so we have the properties of powers. Now the first properties that I want to talk to you about is the properties of the powers. Let's write powers of zero and one. These are very important and you'll come across them every now and again. So let's do a couple of these. The first one is if I take any base and I raise it to the power one. What I'm saying is I'm multiplying something by itself only once. Is this that number itself? And so nothing is going to change. We just still have the number, that number which was in the base, which is the base. The second one is if I take anything and I raise it to the power zero, anything raised to the power zero, we're going to define as being one. And I'm going to say here unless something, and I'm going to say unless x is equal to zero. And that's going to be something very specific and we'll talk about that when we talk about zero to the power zero. So let's assume for this moment that x is any real number, but it's not equal to zero. And if we do raise this to the power zero, we are going to define that to be the number one. And also if we take zero and we raise it to any power, that is always just going to be zero. And I'm going to say unless a is equal to zero. Again, I'm referring here to zero to the power zero. There was referring to zero to the power zero. So let's just have that. I'm going to have zero to the power zero. We're going to say that that's undefined. That is not something that we define in mathematics to be a real number. Now just be careful if you use certain computer languages, if you say zero to the power zero, it's going to give you this property up here. And it's going to tell you that it's one. But in mathematical terms, if we raise anything zero to the power zero, we're going to say that's undefined. But anything other than zero raised to the power zero, we're going to define that to be equal to one. And so those are the properties of powers. The first one we're looking at, and that's the powers involving zero and one. So let's talk about powers involving products. So let's say products. Products of powers. So let's have a look at that one. Now we're going to have two beautiful examples here. We're going to say x to the power a. That's our first power. And then we multiply that by x to the power b. And do notice that we have the same base. That is so important for you to remember. The base is the same. The powers are different, but the base is the same. And in between these two, we have a little multiplication symbol. I'll just denote that with a center dot. And what we're going to say is this is equal to x to the power a plus b. I'm going to add those two exponents to each other. And so let's have a look at a little example of this. So let's put our example down here. I suppose what we should do is just to show that we have a definition. So there's our definition. If we see the same base and we multiply these two powers, I can add those two powers. I can add those two exponents. And let's have a little example. I'm going to take two and I'm going to say two to the power three and I'm going to multiply that by two to the power four. And immediately I'm going to say, well, the bases are the same and I'm multiplying these two powers. And so I can add those exponents. I'm going to say that's three plus four and that equals two to the power seven to the power seven. And let's just check if this is indeed so. Let's just multiply, we're going to write all of these out. So two to the power three, that's going to be two times two times two. Multiplied by, and then four times two times two times two times two. And by our original definition there, we're going to say, well, there's one, two, three, four, five, six, seven. That is two multiplied by itself seven times. And lo and behold, we have the fact that if we have the same base, we can, and we're doing a multiplication there. Most importantly, we're not doing addition on these. We're doing multiplication of these two powers. Then we can add those two exponents. That saves us a lot of time. Now what I just want to say here, what you are not supposed to do, so it's very important to put this in, because we have to think about those. What if we have something like x to the power a, and let's have y, let's make that y to the power b for instance. Can I look and I do that? Can I say that's multiplication and now I can simply write x times y and I'm going to have a plus b? Well, we're going to put a big line through that to say that that is not equal to each other. We can only use this product of powers if we have the same base. And x and y, now it's a variable, that's a variable, it might be that they hold the same values, but what we usually mean by that is that these are not the same numbers. Then I cannot do this. This rule is only there for the bases being exactly the same. And so let's go on to some more exciting examples, because the more of these you do, the better and the easier it gets. So let's have something like 2 to the power 3, let's have something like multiply that by, I'm just going to put dots, multiply that by 4, let's multiply that by x to the power 6, let's multiply that by y to the power 5, let's multiply that by x to the power 4, let's multiply that by y. There's a long expression, and it's actually a single term if you think about it because I don't have addition or subtraction between different terms. It's just one big term, and a single term or more than one term gives me an expression. So let's rewrite this because we can rewrite four in a different way. Let's write 2 to the power 3, and indeed we can write 4 as 2 to the power 2 because 2 times 2 is 4, there's nothing wrong there. And then let's look at the x's, there's an x to the power 6 times an x to the power 4. Certainly the two bases are the same, so I can add those two exponents, 6 plus 4. Here I have y, I have y to the power 5, and y, the two bases are the same. Now if I just have y, I'm going to make use of this property right up here and say that's just the same as saying y to the power 1, because if I take anything to the power 1, it stays that thing. So here I have 5 plus 1 is 6, so 5 plus 1. And for this 2 to the power 3 times 2 to the power 2, the two's are the same, I can add these two, that's going to be 2 to the power 5. And I'm going to have x to the power 10, and I'm going to have y to the power 6. And I can also rewrite this as 32, x to the power 10, y to the power 6, because 2 to the power 5, 2 raised to the power 5, that's a 32. So look at some of these examples. If you do a couple of them, it helps cement your understanding of this idea of defining powers and this idea of the products of powers. So this means we can also talk about the quotient of powers, quotients, the quotient of powers. Now here we defined x to the power a multiplied by x to the power b, but what if we have x to the power a divided by x to the power b? In other words, that's just going to be x to the power a over x to the power b. And now look at these. I have the same basis, there's an x, there's an x, and what I'm now going to do is says that I can subtract a minus b. And that will only be, and usually in mathematics, I want to show you that because you'll see it so often in high mathematics, we say if and only if, and we in mathematics have invented this word IFF. If and only if. Again we're lazy so we have this nice abbreviation. If and only if. Now what I remember is I cannot divide by zero. So how can I get zero in the denominator? Well first of all if b was zero that would not help because any number other than zero raised to the power zero is one. It's definitely not zero. The only way that we can get this if the base itself is zero. So I'm going to say if and only if x is not equal to zero. Because if it's equal to zero and then b is anything but zero, this is going to be zero and I cannot divide by zero. So I just have to remember that when it comes to the quotient as opposed to the powers. So I'm just going to put it to the slightly to the side. They haven't left enough space for myself, but do remember I've got the same base divided by the same base. And that means I can do the subtraction of these exponents a minus b a minus b. I just have to remember that I cannot divide by zero. So now that we've looked at that let's look at another little example. So let's say there's our example. We're going to say something like let's do two to the power five and I'm going to divide that by say two to the power three. Now I see that the bases are the same, which means I can look at these exponents and I can consider subtraction five minus two. And that equals to oh that should be a three. That's two to the power three. So five minus three and five minus three is two. So two to the power two and that equals just four. And let's look at this another way. I've got two times two times two times two times two. And I'm dividing that by two to the power three. So let's do this two times two times two. Now you might remember from school is that if we have this multiplication numerator denominator we can start cancelling out. I'm going to cancel out that two with that two, that two with that two, that two with that two. And so in the denominator I just have left a one. So it's something to divide by one and the numerator I've got two times two. In other words I've got four divided by one and that equals to four as we had in our little example. So let's do another little example. We'll say try and do as many of these as you can. Let's do something like two to the power five. Let's have times three to the power six. Let's do five to the power five and I'm going to divide that all by say three to the power five and I'm going to do five to the power four. So first of all let's look at the numerator. I've got a base of two three and five. Nothing I can do there because these bases are all different. If I look at the denominator three and five for the bases I can do nothing about that. But I do see I have the same base in the numerator there and denominator there, numerator there and denominator there. So let's see what we can do about that. Two I can do nothing with. I'm still left with the two to the power five raised to the power five. But here I can make use of my quotient rule. I'm going to say that's three. The bases are the same. I can do subtraction of these powers. So that's six minus five and I have multiplied by five. The bases are the same. I can do subtraction five minus four. Five minus four. That's going to equal well two to the power five is going to be 31. You can write it whichever way you want. And I'm going to multiply 32. I should say where is my arithmetic? Two to the power five is 32. Three raised to the power one and five raised to the power one. So I've got 32 times three times five. And if you use your calculator please do. You should get four hundred and eighty. Four hundred and eighty. Let's do another example. Let's do something with. Let's do something with some symbols in them. Let's do four. Let's do times x to the power six. Let's do y to the power three. Let's do z to the power two. And we're going to divide this all by let's divide that by two. Let's divide that by x to the power four y to the power three and z. How should we go about that? Well first of all we have a base a base a base a base. They're not the same which is what we mean by these variables. The same goes for the denominator and they're not the same. Nothing we can do that about that. But we see in the numerator denominator we have similar bases. First of all four divided by two is this two. You can do that. But you can also note that four can be written as two to the power two. Two to the power one can be written. Anything to the power one just stays that number. And so I'll have two and two minus one is one and I just have two left. But in this instance it's so easy you can see four divided by two. This is going to be two. Let's look at the x's. Six and a four base and base is the same. So I can do six minus four. I have a y and a y. So I can do three minus three. Three minus three. I have a z and so I'll have two minus that'll also be a one. Two minus one. Whenever you see a variable on its own do remember it's always to the power one. And so four divided by two is this my two that I have there. X to the power two. Y to the power three minus three is zero. And z to the power two minus one is one. And just to simplify all of this I'm going to say two is going to be an x squared. And I do remember that anything to the power zero is just one. So it's multiplied by one and z raised to the power one. Well that's this z. And so I'm left with two x squared z. So please do see if you can get more of these examples. They are great to do. Let's talk about negative powers next. I'm still busy with my properties of these powers. Let's talk about something very important and that's going to be negative powers. So let's have a look at I suppose let's start with an example just to tell you what I'm talking about. Let's have x to the power two and I'm going to divide that by x to the power three. And you'll say well base in base I can do x to the power two minus three but that's negative one so I have x to the power negative one. And now I have to think what do I do if I have negative powers. How do I define that. So let's have a look. I'm going to take any number x and I'm going to raise it to the power negative a and I'm just going to put a little caveat here. Remember x now is just a is going to be a positive number. So when I say negative a I really truly mean a negative number. Now of course later on we can make a a negative number and the negative times the negatives are positive. So this all just turned out to be a positive. But for now let's just make it truly a negative number. And we're going to say that that is equal to the following. That's going to be one over x to the power a. That's how we define this x raised to the negative number. That's one over x to the power a. And once again I cannot divide by zero. And so I just have to have this little thing if and only if x is not equal to zero. If x is equal to zero zero to the power anything that's not zero is going to be zero. Zero to the power anything is zero. Except if it's zero to the power zero which is not defined remember. So I cannot divide by zero so I can't have zero to the power negative a. That's not going to work for me. But I do want you to remember from school just remember something called the reciprocal. So if I have a number four its reciprocal is going to be one over four. This was actually four over one and I just flip those around. Such as that when I multiply these I'm going to get equal I'm going to get this equal to one. That's something a bit more advanced that we can talk about at a later stage. But this reciprocal so that if I have a look at this once again that this is just going to be one over x. So I can take any number any number raise it to the power negative one. If it's one negative one that's going to be the reciprocal of that number. But now we can talk about what if it's negative two negative three etc. That is something else. I just want to remind you of that. Now let's put this as a definition because that's quite important. This is how we define negative powers. So I want you to remember that extra raise to the power negative a. Now a is a positive number times a negative that's truly a negative. We're going to write it as such. So let's look at some more examples just to make this clear. Let's solve two to the power negative three. How would I go about that because you know if we just look at our definition how do I multiply two by itself negative three times that doesn't make sense. So we're going to make use of this definition of ours for negative powers. Well we're going to say well I can just rewrite this as two to the power three. But I'm just going to put that in the denominator. Now it becomes positive three and I just put that in the denominator. That's going to be one over eight and that's going to be 0.125 0.125. And so that is such an important thing for us now how do we work with this example where we have a negative in our exponent a negative in our power. Well we do have a definition for that. We bring it down into the denominator such that we can still work out a value a solution to this problem. And that means we can also have something like this. What if I were to write something like this one over. Let's make it x to the power negative three. How am I going to solve that? I have x to the power negative three over there. And what I have to do is just to think about this at least in reverse order. That if I have x to the power summing I can rewrite this if this is my denominator rewrite it in this way. So I'm just going in this direction. So that is going to be well take whatever is in the denominator that is whatever is in the denominator. And I want to bring it into the numerator just you know bring it over to the side but I do have to take that a and a multiplied by negative one. So I'm going to multiply this by negative one. And so negative three times negative one well that's positive three and that's going to be x to the power three. So if I have that negative down here in the denominator that is nothing other than x raised to the power positive three. So this is actually I think a very powerful example that I want you to really think about and really concentrate on because that allows us to solve something like this what if I have this very weird fraction one divided by a quarter one divided by one over four. Well let's think about this let's think it through my numerator let's keep that one but I have this one over four there's a four can I not rewrite that as four to the power negative one. Let's have a look at this what is four to the power negative one well that is just going to be well bring that down into the denominator and one over that four to the power one that is just a quarter. So that's another way to write a quarter and there we have a quarter there's another way to rewrite it and now I'm going to make use of this example just putting all of my definition in reverse order so I'm going to have whatever was in my denominator and I'm going to take that power and I'm going to multiply it by negative one negative one times negative one's positive one and that equals this four a very long way just to write four. So take some time and look at these examples they are very very important when it comes to negative powers let's have a look at the next thing on our agenda and that's going to be the power of powers which sounds a little odd but I like to not use the term exponents here you'll see these written as exponents etc I like to say powers are powers powers of a power and let's define that I'm immediately going to write x to the power a and all of that is to the power b how do we define that now this is not x to the power a and then multiply by x to the power b that's what we had before this is x to the power a raised to the power b well that's just going to be x to the power a times b x to the power a times b let's put that as a definition very important and we do have to distinguish that from x to the power a times x to the power b that's not what we're saying we're saying take x to the power a and multiply it by itself b times multiply it by itself b times so the easiest way to consider this is to look at an example let's have two to the power three all of this raised to the power two and think about this I'm just taking this and I'm multiplying it by itself twice so that will be two to the power three times two to the power three and now we back to one of those original laws that we have this is the same base and I can just add these two so that's going to be two to the power six but notice that would just be three times two that is two to the power three times two let's put three times two and that's two to the power six so these can be very confusing it's not that I'm multiplying two to the power three and then I have two to the power four or something like that and I have to add the exponents no this is this whole value which if you think two to the power three is eight eight to the power two so two to the power three is actually my base and I'm multiplying that by itself twice and we can then multiply these two powers let's just do another simple little example let's have something like two to the power three I'm going to raise that to the power four well that's just going to be two to the power three times four and that's two to the power twelve that's a very simple concept but we can get a little bit confused by this this is rewrite this that's going to be two to the power three times two to the power three times two to the power three times two to the power three now these are all the bases are the same so I have to add those that's going to be two to the power three plus three plus three plus three and that's two to the power twelve let's bring those negatives in when it comes to an example let's have something like two raised to the power negative three and I'm going to raise that to the power two well that'll be just two to the power negative three times two to the power negative three or I can just remember that I can multiply these that's going to be two to the power negative six and that's going to be one over two to the power six there's so many ways that we can write this and 2 to the power of 6 is of course 64 so that's 1 over 64 and again you can rewrite this if you divide that with a calculator hopefully you get 0.015625 check that in your calculator make sure that that is correct so here we can still make use of this power of you know power of powers in my power yeah I'm doing multiplication I'm doing powers of my power one more example these are just so much fun and they're so important to work with let's bring in something like this let's take 3 I'm going to say x to the power 2 so x squared and y cubed y to the power 3 and let's take all of this to the power 2 well that's going to be 3 to the power 2 now remember this is 1 so it's 1 times 2 is 2 x to the power 2 times 2 is 4 and then y 3 times 2 is 6 and there we go so we can rewrite this 9 x to the power 4 y to the power 6 very simple to do if you think about it let's go on to power let's call it power roots there's so many names look in different textbooks you'll find different names and so one that I immediately want to show you is let's go for this let's say x to the power a divided by b now that is something very special in our power we have a fraction we have a fraction and that is actually called I'm going to call this whole thing a radical let's just call this we have here as a radical and we have to think about what do we do what do we do with us how do we define all of this and actual fact I think let's go there we've got different names for these this a in the numerator of the power we're going to call this the radicand and the denominator this b we're going to call this the index so do look out for different names and different textbooks but let's stick with these for now and I'm going to say that this is this is a a radical we have a number raised to the power of a fraction so let's have a look at an example and let's do something very familiar might have seen us before x to the power one half and you might have noticed that that is rewritten as a square root and we put a little two there but usually we'll just leave that two and then we have x to the power one so what happens to this index it goes right there when it comes to the root and we take the numerator there and that stays the power of whatever we have there and that's usually just written as the square root of x now do remember x must be non-negative for us to still have a real number when we define the square root of of um um a number let's have another little example I'm going to have x to the power two thirds and so what I'm going to do is I'm going to take my x my base I'm going to keep my numerator there and then the three the denominator is going to go into this root so that's what we're going to call a cube root of x squared so finally we can have let's write this this is still going to be my x I'm going to keep this raised to the power a and then I'm going to have the root right there I'm going to put the index right there so I'm keeping my reticant there's my base and my index goes there let's put that as a definition and by definition here I just mean this is the things that we really have to to remember now one thing we have to talk about I suppose as an example we can do that is if we have the same if we look at this power that's a fraction if those things are the same so let's do a very easy example I'm going to have the square root of x times the square root of y what am I writing here I'm saying x to the power a half times y to the power a half now I don't want to confuse you before remember when we could add these halves to each other but then we need these two to be exactly the same so don't make that mistake but what we can do here is we can rewrite this as the square root of x times y in other words I'm going to have x times y to the power a half so that means these two are the same and I'm multiplying then I can go ahead and do this so again a very very powerful example now one of those that we have to remember let's do it as an example with numbers in it so let's have two to the power half times three to the power a half I remember that's the square root of two times the square root of three and I can rewrite this as the square root of two times three which is going to be the square root of six just one of those things I think that is this useful for us to remember to get used to let's build on that let's do another little example this one might be a bit more difficult let's take x to the power two thirds and I'm going to take y to the power let's make it four thirds now what do we do these two are not the same those two are not the same can we make sense of this well yes we can rewrite this we can rewrite this in a number of ways actually now remember here we can take x and we can keep that ready can that too and this is going to be a cube root and we're going to multiply that by another cube root there's the three so I'm going to put a cube root there y to the power four we've got the same roots here cube root cube root that is going to just be the same as taking the cube root of x square times y to the power four now let's just think about that just a little bit here can I make this look like that so I do remember that these have to be the same for me and for me in order you know to use these examples and let's have that I can rewrite this as x to the power two to the power a third now look at these two they are exactly the same I do remember what I do have up here I can do a times b a times b two times a third is two thirds nothing wrong there and here I can take y to the power four to the power a third four times a third is four thirds no problem there and now I have a third and a third can you see there was a half and a half now I can make use of this so that a third which means it is a cube root and I'm left with anything to the power one is just that thing so I still have my x square there and I have my y to the power four there exactly as we had up there so I do remember if I can you know if I can rewrite this so that I have these same radicals there's to the power one over three to the power one over three which is nothing other than a cube root I can make use you know of these the the law that we're trying to develop here there's my x squared there's my y to the power four and even though these two don't look like each other they are actually the same and I can make use of this very idea and so what do we have here what is the law that we're trying to develop here we're trying to understand what we can do when these indices are the same remember the denominator is my index there was my index two and two they're the same there's my index three and three they are the same and I might have to do some extra work to remember the two thirds is two times a third four thirds is four times a third and I have the exact same and now we have three and three we have three and three we're thinking about this idea what do we do when this index when the indices are the same well then we can have all of these examples let's do one more example and this time we're just going to do division so what if we had two to the power a third and we're going to divide that by it's divided that by three to the power a third now first of all we can rewrite this of course that's two to the power a third divided by three to the power a third now remember this is to the power third which we have mathematical notation for we just write that that's the cube root of two divided by the cube root of three and we can make use of this rule where as same as with multiplication we can put this under the same cube root so that'd be two divided by three remember that's going to be the same as saying two over three all to the power a third it means I'm applying this to the power third to both the numerator and the denominator and so these are actually quite easy to do and they are exactly the same thing as we did with multiplication what I do want you to remember though is the following what you cannot do if for instance we have let's do let's do the a th root of x plus or minus the a th root of y that is not equal to if I do the a th root of x plus or minus y that this rule only applies to multiplication and to division it does not apply to addition or subtraction only multiplication and division so let's do this example let's do a to the power four over three and we're going to divide that by b let's divide that by seven to the power seven over three and you might think well now I'm stuck it's not as if I have a third and a third and it's division or multiplication yet is division and so I think I want to make use of this rule but I do not have the same power there's four over three and three over three but what I do want you to recognize is the following that I can write this as a to the power four to the power a third and let's bring our ruler in and let's divide this by b to the power seven to the power a third and then we remember our law where we had exactly as we had appeared we had something to the power third something to the power third something to the power third something to the power third so we can make use of this law but I do want you to remember if I have a to the power m to the power n it's just going to be a to the power m times n and so four times a third is four third and seven times a third is seven thirds but now I have a power a third a third so I can rewrite this as a to the power four divided by b to the power seven all to the power third or if I want to rewrite that as a cube root that's going to be a to the power four divided by b to the power seven and no matter which way around I write it we try we mean the same thing so don't be fooled when these powers are not exactly the same look at this denominate here look at the index here the index it's the same and hence I can make use of this law that we learned about before so that I have these same powers please do remember that it is not applicable when I have addition or subtraction and then I want you to remember a couple of these they're just useful later on we're going to learn about factoring etc and these might make slightly more sense but it's just worthwhile to remember these if I have something like m and I have the ath root of say x plus let's say plus or minus n and the ath root say of x and you can see this is the same the ath root of x the ath root of x and what I can then just do is bring that m plus or minus n out on its own and what I'm actually doing is taking these out as a common factor but we will learn about that later and that is going to be the ath root of x so it's just one of common things that you might want to remember the next common one I want you to remember is if I have the square root of a constant times the square root of another constant now that's going to be the square root of c times c and I'm making use of this exact same law that we had up here that is going to equal the square root of c squared and remember I'm writing nothing other than here the c to the power 2 over 2 that's what I'm writing here and that's just going to be c because 2 divided by 2 is just 1 and c to the power 1 but what I do have to remember that it has to be the absolute value of c remember we can't take the square root of a negative number when we just want to deal with real numbers we don't consider the complex numbers so we are talking about the solution here if I take that the square root of c squared that I'm talking about the absolute value of c or the positive version of c one more that I want you to just to note if I have the ath power of x remember that I can just rewrite this as x1 over a that's what we've been dealing with all along so that one you'll definitely know and then if I have x to the power b a remember that's this x to the power b over a my radicand and my index it's just different ways of writing it and we've been doing it all we've been doing it there all along now the next big topic I want to talk to you about is exponential expressions now what do I mean by that up until now we've taken some unknown number x and we've raised it to a certain power but what if this becomes the power what if I have something like a to the power x now we're going to say that that is going to be our variable this is going to be a constant so it's a constant raised to the power variable here I had a variable raised to the power a constant so this is the other way around the most important one that we have actually is called the exponential function and you'll see some textbooks talk about the exponential function and that is where this constant is oil is number e and we raise that to the power x we're going to call that the exponential function later on we'll talk about the difference between expressions and functions but for now just remember when we use the term exponential function we're usually referring to this but in more general terms I don't want this constant to be just the number e oil is constant or identity I can think of this a being any constant but I'm raising it to the power of a variable and so how do I deal with these now there are some properties to these and these are so important properties there goes my pencil that is so important so let's have a couple of these I'm going to have a to the power x and I'm going to multiply that by a to the power of y so what I'm saying this is a constant this is a constant that's a variable that's a variable the important here my two constants are the same and I can simply keep them there and I'm can have x plus y and so that shouldn't be too surprising to you again if I have a to the power x and I divide that by a to the power of y my bases are the same so I can stick with my base which is a constant now and I just take x minus y I do have to remember though that I cannot divide by zero so I'm going to say if and only if a is not equal to zero I cannot divide by zero what about something a to the power x to the power y well this is going to be a to the power x times y and you think that well that's nothing that we haven't really seen before nothing that we haven't really seen before what about if I have a to the power x and I'm going to have b to the power x now these are not the same but these two powers that I have these two exponents that I have they are exactly the same well what I can do here is say a b and I raise those to the power x let's see an example just to make sure that this last property does indeed hold at least by one example three to the power four and I'm going to multiply that by two to the power four I have the fact that I have exactly the same values up there and I'm going to say well I can do this that's a three times a two which is a six to the power four and use your calculator I think that should be about one two nine six one thousand two hundred and ninety six but think about this three to the power four check this on your calculator as well that's 81 two to the power four that's 16 and 81 times 16 learn behold that should also be 1296 so check on that check on the fact that we have at least this one example where this would hold so we've had these properties of the exponents I really want you to look at them and you know do remember these they come in so handy when we work with when we work with all kinds of data analysis next important topic that we have spoken about before but I want to expand on that a little bit here I'm going to talk about logarithms and so by definition let's see if we can remember what that is we have the log and we have a base we're going to call that base b of x equals some value y what do I mean by that well that's going to hold if and only if do you like to use that term in mathematics we do so if and only if we have that b to the power y equals x what are we saying with us we're saying if we take if we think about a logarithm we're asking what must I raise the base by to get this value what must I raise b by to get x what must I raise b by well by y to get x and let's put a little green arrow there because I always want you to think along these terms that is such an important concept and so let's put a little green arrow there this is what we mean by taking the log of a number we're saying this base of the log to what must I raise this base by to give me this value that I'm taking the logarithm of well the solution to that is why I must take the base and multiply by this at least raise it to the power this number to give me this value and again as a little example to remember that if I take 10 to the power three that's 10 times 10 times 10 is going to be three zeros there that is going to be a thousand so I can think what is the log base 10 of 1000 well I'm asking what must I raise 10 by to get a thousand well I must raise 10 by three so the log base 10 of a thousand is going to equal three now a very important thing if we think about the base what values can the base take well for us here we just want to keep the base two three four and we're going to go up but we can also have other bases a nice base for us would be e when the base is e that is going to be the natural logarithm we're going to call that the natural log so we're going to write log base e of some value a so that's an e and we have different notation for that remember we just write ln of a that will be the same as log base e we have this ln for natural log many computer languages we just write log and by log we mean base e or the natural log now let's talk about properties properties of the logarithm very very important now the first property I want to talk about is if I have log base any base let's just keep that base b and I've got the multiplication of two values x and y now it is so important in data science I have a multiplication problem here it would be so much easier to turn this into a if I just looked at that multiplication turn that into an addition problem while the logarithm helps me turn in a multiplication problem into an addition problem because this is equal to the log base b of x plus the log base b of y so I do have to have the same base base and base and I have multiplication there I could also think of the log base b of something like x divided by y well that's going to be the log base b of x minus the log base b that's got to stay the same of y these are just two such important and aspects of algebra and we use them over and over and over again so my my base stays the same and I've got to have multiplication or division there that is the only way that it's going to work or if you see this the bases are the same and I'm doing addition you can rewrite it as such if I have the bases are the same and I have subtraction between two logs then I can rewrite it as such but I cannot have x plus y here and x minus y there that's usually where the confusion lies it's got to be multiplication here and I'm taking the logarithm of that multiplication I turn that into the addition of two separate logs I turn division into the subtraction of two separate logs the next one is let's have log I'm going to have base b and let's have a different number I'm going to have a to the power c so I'm taking the log base b of a to the power c now what I can do there in that instance is take this exponent and I can bring it to the front that is going to equal c times the log base b of a another very very important one and then the last one I want to show you the last one of these laws if I take a log and I have that to some base b of a I can change this by saying the log base now let's make a different base I'm going to make base c of a divided by the log base c of b and that's called the change of bases maybe I don't like the base b I want to change it to the base c so I take my number that I'm taking the log of there it is but I'm changing this to the log base c but now I have to divide it by taking the log with this new base but of the old base right there so those are names apologies for it being on two separate pages but those are the four very important properties that you have to remember when it comes to logs and again as with the exponents these are so so important for you to remember so let's do a couple of examples just to remind ourselves I'm going to have the log base let's make it base 10 and I have the number three times the number four well that's going to be log base I take keep that same base there and I have the log three plus the log base 10 of four so let's have an example of the log base let's keep it base 10 of three over four and that is going to be the log base 10 of three minus the log base 10 of four I'm taking this fraction I'm taking that product and I can rewrite this as subtraction and addition let's have another example I'm going to take log base let's take it log base 10 of let's make it a hundred to the power three now I do remember I can bring the three to the front so that's going to be three times the log base 10 of 100 and the last example let's do something like the log let's make it log base seven of five but I want to rewrite that say as log base 10 so that's going to be log base 10 I keep my five and I'm going to divide that by log base 10 so those have to be the same of my old base seven so I've changed from a base seven to a base 10 by doing this five divide by seven but it's a log base 10 of five divided by the log base 10 of seven that is this change of basis and there you go what you see now is the most important part of taking powers considering powers and considering exponents and lastly again considering logarithms you have to practice these if you are taking the course do remember to do all those homework problems for your credits