 Hi, I'm Zor. Welcome to Unizor Education. Today, we will consider another type of a function. It's called exponential function, and let me start from the definition, and then I will go into the explanation of what it actually means. So the definition of the exponential function is this. So x is an argument, y is the value of the function, and a is any real number. Well, actually, zero is not an interesting one. Let's forget about the zero. So a is positive. Now, x is all real numbers. Well, it's fine to define something, but then I have to explain what I really mean, right? All right, let me start from a very simple case. If x is natural number, usually when we are talking about natural numbers, we try to use some letters like m, n, or k, or whatever. So let's talk about a to the power of n, where n is natural number. Actually, we do know what that means. By definition, this is a times a, etc., times a, n times. Now, that's easy. Okay, so the definition of the exponential function for natural exponent, this is called exponent, a is the base, x is exponent. So for natural number as an exponent, it's very easy to define. That's exactly how we did it. What if I want something like 2 to the power of 1 third? That's not as intuitively, at least looking at this particular definition, right? So we have to really explain what we mean. And that's exactly what this particular introduction into the exponential function is about. So this lecture is about to what exactly we mean when the exponent is basically any real number. And I will gradually build this particular explanation step by step, basically increasing the complexity of that thing. So as a result, you will know what a to the power of x means for any real number x. Okay, let's start from the beginning now. And let's expand our obvious definition of exponential function for natural exponents. Let's expand it towards more and more different numbers. Now, before doing that, I would like actually to point two very important properties of this particular expression where a is positive real number and n is natural number. So I would like to mention a couple of very important properties because I would like these properties to be preserved for all other types of exponent, rational, irrational, negative, zero, et cetera, et cetera. Because if we want to reasonably define the exponent for non-natural numbers, like rational, for instance, we would like certain properties, good properties which we can observe with natural exponent. We would like these properties to be preserved for all other types of arguments, types of exponents. So what are these important properties which I would like to preserve for all other types of exponent? Here is the most important one. If I have a sum of two different natural numbers as an exponent, this is, let's just think about what it is. This is a to the m plus n power, which means it's a times a, et cetera, times a m plus n times, right? Now, if this is m plus n times where m and n are natural numbers, then I have m times and n times, correct? Now, this is a to the mth power and this is a to the nth power and they are multiplied by each other. So this is a very important property of natural exponents. Just as an example, 2 to the power of, let's say, 6 is what? 64, right? Now, 2 to the power of 4 is 16 and 2 to the power of 2 is 4. 2 to the power of 6 is equal to 2 power of 4 times 2 power of 2 because 6 is 4 plus 2. 16 times 4 is 64. What did I put here? I'm sorry. I mean that, right? So 16 times 4 is 64. So this is obviously preserved. I mean, this is not the proof. Proof I did before. This is just an illustration that this is a very important property of exponents. Now, if I am a reasonable person and I'm trying to expand the definition of the exponent to other non-natural numbers, I would like to define it in some way where this particular property is preserved. I mean, otherwise, it looks like for some numbers the property does exist and for some other it does not exist. I would not consider it reasonable. I mean, probably I could do some crazy definition like that, but why? I really should try to define the function 8 to the power of anything in such a way that this important property is preserved. By the way, this important property can be just slightly expanded towards more than one numbers together. So if you have m1 plus m2 plus m3 plus et cetera plus mk, then it obviously can be written as 8 to the power of 1 times 8 to the power of 2 et cetera, 8 to the power mk. The proof can be... I mean, rigorous proof can be done by induction. Non-rigorous proof is just obvious because this is A times A times A, this number of times, and we just divide this product of m1 plus m2 plus et cetera times A. We divide it into groups m1 and 2 mk times product of A by itself. So this is obvious, and one of the consequence from this is if all m1 and 2 and 3 et cetera are equal to each other and they are equal to some number m, so I will have A to the power of m plus m plus m plus mk times, right, so it's k times m. And this is what? It's A times to the m's power, 1 to 3k times, which is this. So this is yet another property which is obviously a consequence of this one. And it's all based on the same property that the addition of exponents results in the product of the exponential functions. All right, let's just remember this and this. So let me just put it on the top. So we will have a reference to these two things. A to the power m times k is equal A to the m to the power of k. Incidentally, it's also equal to A to the power of k to the power of m because m and k are commuting to each other. All right, so these are properties and we would like to preserve them for all other types of exponents, not necessarily the natural. Now, the first is 0. What is m is equal to 0? Well, in theory, I can have some philosophical discussion about if A to the power of m is A multiplied by itself m times, then A to the power of 0 means A multiplied by itself 0 times. But what is this? 0 times multiplication. I don't know what it is. But let's go into this rule and let's try to basically use this rule to find out what is a reasonable value for A to the power of 0 and then just define that A to the power of 0 is this reasonable value. That's how people do. So our definition must be based on some logic and here is the logic. A to the power of n is equal to A to the power of n plus 0, right? Because n is equal to n plus 0. Now, using this rule, n plus 0, I will just use it formally. It's A to the power of n times A to the power of 0. You would like this rule to be preserved and what follows from here? A to the n's. This is A to the n's times something. Now, what is this something can be? Only 1, because only multiplication by 1 leaves A to the power of n without any change. So from this equation, if you wish, we derive that A to the power of 0, which we don't really know what it is because it's not defined. But it must be equal to 0, sorry, to 1 just to be a reasonable definition. Because if this definition is not like this, if it's something else, if it's some other number than 1, then if we add 0 to our repertoire of different exponents, we will not have this property preserved. So we must define A to the power of 0 as equal to 1 unconditionally, regardless of the value of the base A, by the way. So no matter what the base is, if you use A to the power of 0, it must be equal to 1. Otherwise, this will not be preserved. So that's the definition. We came up with the definition. It's not a proof. Let me just point it again. It's not a proof that A to the power of 0 is 1. It's a logic which is behind the definition. So A to the power of 0 is equal to 1 by definition, but the logic behind this is this one because we don't want to break this particular property. OK, fine. So we have defined A to the power of 0 is always 1 regardless of the base A. Next. Next, let's expand to negative numbers, integer, negative integer numbers. A to the power of, I'll put it minus m, assuming that m is a natural number, 1, 2, 3, et cetera. So it will be A to the power of minus 1 or minus 2 or minus 300, et cetera. So how this can be defined? Well, let's do this little operation. A to the power of n times A to the power of minus a n. According to this, it's this, right? If you multiply these, you add the exponents. Now, this is obviously A to the power of 0 and plus minus n, right? And we already know this is 1. Now, what follows from this is the following. That A to the power of minus m, this one, must be equal to 1 over A to the power of n. Let's disregard the phone. OK. So basically, this is something which is, again, a definition. But it's a reasonable definition primarily because we want this property to be preserved. If we don't do this, if we don't define it as such, then I probably would not have this particular equology. So this property would not be preserved. To preserve the property, I have to define A to the power of minus n as 1 over A to the power of n. Again, this is a definition, but it's a reasonable definition because the property is preserved. So we have defined A to some power in many different cases right now. We defined it for natural numbers as the multiplication of A by itself. Certain number of times, we defined it for 0, which is always unconditionally equal to 1 regardless of the base A. And we have defined it for negative integers. So A to the power of minus 2, let's say, this is 1 over A to the power of 2. By the way, here I explicitly use that A is not equal to 0. Now, the fact that it's positive is not maybe used so far, but it will be used in the future. Because I would like to expand to, let's say, square roots of something. And we don't really know how to... I mean, we do know that square roots of some real numbers can be expanded into complex numbers, but we're not talking about complex right now. Only real numbers. So we are dealing with real numbers, and that's why A should be always positive. Okay, so we've done that. And let's now go into a little bit more complex case. Okay. So what after positive and negative and zero integer numbers? Well, we have to expand it to rational numbers, all right? So let's just think about rational numbers and what is a reasonable definition for exponent being a rational number. Let me start with the following. What is A? I can always put A as A to the first degree, first power of 1, right? Because it's A times itself one time, which is A. Now, 1 can always be presented as Q divided by Q, where Q is integer positive number, natural number, right? So Q is natural number. So Q over Q is always one. Now, what is Q over Q? I can always present it as A over in the power of 1 over Q plus 1 over Q plus et cetera, plus 1 over Q, where I add 1 over Q Q times, right? It is the same thing, like 3 third is 1 third plus 1 third plus 1 third. So Q Qs is 1 Qs plus 1 Qs plus 1 Qs et cetera, Q times. Now, I will use this property that edging exponents results in the product of exponential function. So it's A to the power of 1 over Q times A to the power of 1 over Q times et cetera, times A to the power of 1 Q plus its multiplication. 1, 2, 3, 4, Q times, right? This is Q times we add 1 Qs to itself. So this is Q times multiplying A to the power of 1 Qs by itself, which is equal to A to the power of 1 over Q to the power of Q. Let's recall that if you have B to the power of, let's say, M equals to A, what does it mean? It means that B is equal to root, M's root of A, right? This is definition. What is M's root of A? It's such a number B, which B increased into the M's power results in B, right? That's the definition. So basically, what I can say in this particular case, A to the power of 1 over Q is basically my B in this case. So 1 A to the power of 1 over Q is equal to Q's root of A. So we all know what Q's root of A is. It's defined as the number being raised into the power of Q gives A. So that's what A to the power of 1 Qs is by definition. Again, this is a definition, not a proof. The definition of A to the power of 1 Qs, where Q is a natural number, is Q's root of A. It's a reasonable definition because we have this basically the same property is preserved. So that's it. And the only thing which we have, well, not the only thing, just a little bit more, we have to define A to the power of P over Q, where P and Q are natural numbers, positive integer numbers. What is this? Well, you can always consider this as A to the power of 1 Q to the power of P, right? Using this property. This is P over Q is P times 1 Qs. So it's A to the power of 1 Q raised to the power of P. Now, we know what this is. So it's Q's root of A. So that's to the P degree. And from this, we very easily deduce that this is equal to the A to the power of P as the first, then being rooted by Q's degree, which is the same thing basically as you can do it conversely. This is A to the power of P to the power of 1 Qs. This is exactly the same thing. The Q's root of A to the power of P. So no matter how we use, as I said, these are symmetrical things. So no matter how we do it, we derive with the same conclusion that A to the power of P over Q, where P and Q are natural numbers, is the Q's root of A to the power of P. Okay. It's almost done with rational numbers because all we have to do is to add negative numbers. We already know how negativity actually affects. So if this is a negative number, a negative rational number, let's say it's A to the power of minus P over Q, where P and Q are natural. So this is a negative natural number, right? Now, we all know what negativity actually does. So remember, A to the power of minus M is equal to 1 over A to the power of M. We would like to preserve it, right? So to preserve it, all we have to do is define a negative rational power exponent as 1 over Q's root of A to the power of P. That's it. It's a definition which I have derived from everything which I've had before. Power of P over Q. I mean, I can do it in two steps actually. This is minus 1 times P over Q, which is A to the power of P over Q minus 1. Minus 1 gives me 1 over and the A to the power of P over Q gives me Q's root of A to the power of P. So no matter how we do it, this is a definition for negative rational numbers. What's left is only the irrational numbers. And here is where robustness and rigidity, rigorousness of all this theory actually breaks, I would say, a little bit. You see, to properly define real numbers, we need a theory of limits which is addressed in these lectures but a little later. So that's why I'm using semi-rigorous description of real numbers as being approximated, however precisely, by rational numbers. So I can always say that if rational number P over Q is an approximation for the real number R, then A to the power of P over Q is approximation of A to the power of real number R. The more precisely we define this, the more precisely this is actually defined. So the property, since this property is preserved for rational exponents, using this approximation behind which is actually the theory of limits stands. But anyway, without mentioning the theory of limits, let's just think about approximation. Since the approximate value observes this particular quality, it has this particular characteristic, then any irrational, any real number therefore has the same property and it's defined this particular way. So I might or might not actually address this more rigorously with real numbers. Quite frankly, I don't see the need for this. But for instance, if you want to know what E is, let's say 2 to the power of square root of 2, which is irrational number. All I can suggest you is use 2 to the power of 1.4 is an approximation, or 2 to the power of 1.41 is an approximation, etc. So I'm using these decimals, but at the same time I obviously can use 1410s or 14100s, which is the same thing. And it adheres to our definition of rational exponent. So these are approximations. And obviously I can say that if this, let me just clear it out, if I want to know for instance what is 10 to the square root of 2 plus square root of 3 exponent, I can always say that this is 10 to the power of square root of 2 times 10 to the power of square root of 3, which in turn approximately equal to 10 to the power of 141100s times 10 to the power of, I don't remember exactly, it's something like 1.7. So it's 1772 or something like this. So no matter what it is, it's more or less approximating. Let's stop here, let's not go any deeper, which I don't really think there is a need. What is actually important is to notice that this property, if you add two exponents, even irrational exponents, it still can be broken into the product of two components, it's still preserved. So the definition using the approximation with rational exponents really works for irrational as well. So you have the right to have this particular equality, even if 10 to the power of irrational number is not very rigorously defined, but you can definitely consider it as the one which we have basically defined. But now let's think about this differently. This is equal to 10, what is square root of 2? We already know that this is 2 to the power of 1 half, right? Because this is the square root, 2 is the root of 2, right? And this is 10 to the power of 2 to the power of, sorry, 3 to the power of 2. Now what's very important in this case are these parentheses, because obviously this is not the same as this. You see, I put parentheses differently here, and it's a completely different number, because 10 to the power of 2 is 100. 100 to the power of 1 half is the root of the second degree of 100, which is 10. So this is 10. Now this, or this, is definitely not 10. 10 to the power of square root of 2 is 10 to the power of 1.4 something, which is greater than 10, obviously. So these parentheses are very important and you have to really be very careful when you are presented with the problem and the parentheses are at certain place. You do not really freely change these parentheses. It's not like you have the associative law. If you remember, if you have the same kind of operations like 2 times 3 times 5, you can put parentheses here, or you can put parentheses here, and they are equal to each other. This is commutative law. This is not a commutative law. So the power is not a commutative operation. Because if you put parentheses here, 3 to the fifth degree, I can't even count how much it is, but it's definitely not equal to 2 to the third degree to the fifth degree. That's not the case. Now, why? Let's just think about why. Product is symmetrical operation. A, B, B, A, it's exactly the same thing. Raising to a power is not symmetrical, because one is a base and another is an exponent, which do have a relationship, but they're not symmetrical relationship. 2 to the third degree is 8, 3 to the second degree is 9. They're not equal to each other. So the operation of raising into power is not commutative and definitely not associative. So it's a completely different animals. You cannot really deal with them the same way as with products or summation operation. Nevertheless, the purpose of this lecture was to define the operation of raising into some power raising a positive number. To define this raising into power for different exponents, they may be either natural numbers, they can be zero, can be negative integer numbers, can be a rational number or negative rational number, positive or negative, or it can be irrational number, which means we cover completely all the sets, all the set of real numbers. So that's why we can talk about this function where A is greater than zero and X is any real number as a function which has a domain of all real numbers. Now, the properties of this function will be discussed in the next lecture. So for today, I want to just do the introduction. Notes are on this side as always. Try to review them again. The purpose was to define and again, don't forget this is a definition of A to the power of something. It's not the proof that A to the power of zero, let's say, equals to one. It's a definition but the logic was needed to basically demonstrate that the properties of thus defined exponents are the same as the properties when the exponent is a natural number. So that's very important. That's it. Thank you very much for today and don't forget that the next lecture in this topic will be dedicated to the properties of the function which we can consider right now as fully defined for all real numbers X. Thank you very much.