 Hi, I'm Zor. Welcome to a new Zor education. I will continue talking about exponential functions. The previous lecture was dedicated to basically definitions of what is an exponent is. Well, obviously, it's very easy to define for natural number as an exponent. So if you have where x is a natural number, then it's just multiplication of a by itself and times. But then I expanded this definition to rational number, zero negative numbers, irrational numbers, basically all based on one very important property of exponent. That if you have a product of two elements, each one of them is some base and some exponent, then it's equal to this. So the product of exponents is exponent of a sum with the same base, obviously. Now, this is a fundamental property of exponential functions, and all definitions in the previous lecture were based on this. Now, I will go into properties of exponential functions, now just to illustrate these properties, I would like you to have in mind a graph of this particular function. I will explain all the details about graph in the next lecture. This lecture is about properties, but the graph is based on the property, and that's why I would like you just to keep in mind how the general picture actually looks like. Now, this function is defined for all positive a, and we have to define it in two different categories, a greater than one and a less than one. So for a greater than one, the graph looks like this. It's increasing from zero to plus infinity, and in this particular case, it's decreasing. Now, I draw these graphs without basically any foundation. The foundation is the properties which have gone to talk about this lecture, and the graph will be more explained in more details in the next lecture. However, it's easy to talk about properties if you have these pictures in mind because I have it in my mind, that's why I would like you to have it. For greater than one base, it's increasing for less than one, it's decreasing. Now, having this picture in mind, now I will go to all the properties. Again, these are the properties which are in the foundation of graphing the function. So that's why it's very important. Okay. Property number one, exponential function is always positive. Now, I will try to prove as rigorously as possible all these properties. So just bear with me. Now, why a to the power of x is positive? Well, let's consider again the definition, starting from the x-natural. So a to the power of n is a times a, etc. a is always a positive number. So if you multiply positive number by positive number, it will be positive number. So for any natural n, it's positive. Now, for negative integer where n is natural, so it's minus n, this is by definition, one over a to the power of n inverse. Now, inverse to positive number is still positive. Inverse to three is one-third. Inverse to a million is one millions, still positive number. So for all integers, positive and negative, the a to the power of x is positive. Now, how about rational numbers? Well, again, what is a to the power of p over q, where p and q are two natural numbers? Well, again, by definition, this is the q's root of a to the power of p. Now, a to the power of p is positive. Now, q's root also positive. Basically, it's a definition of the root. The root is always positive from the positive number. We are not talking about negative numbers, complex numbers, et cetera. We're talking about positive numbers, and root of the positive number is always the positive number, which being raised into the q's, power will give a to the power of p. And obviously with the negative rational numbers, where p and q are natural, this is again by definition. And since this is positive, the inverse is also positive. Now, just once I will address the irrational numbers, and then I will not talk about the rational numbers at all, because I cannot really make it absolutely rigorously. So I was talking about exponential function with x being an irrational number as a limit case when we are approximating this irrational number with rational numbers. So if x can be approximated with a sequence of rational numbers, then a to the power x would be a limit of a to the power of all these elements of the sequence. So basically, I don't want to go any deeper in this because again, we are not really on the level where I can make it absolutely rigorously. So just consider the irrational number as a limit case for rational numbers, and that will probably be enough to talk about, well, the proof that a to the power of x would be all this positive regardless of x. Positive x, negative x, anything. How about zero? How about a to the power of zero? Well, this is always one by definition. Again, that's from the previous lecture and that's basically is based again on the main property of the exponential function. We must have a to the power of zero equals to one to satisfy this and again, that was addressed in the previous lecture. So if exponent itself is equal to zero, then the exponential function is equal to always one. Okay, how about base? What if base is equal to zero? What happens then? Well, then exponential function is always equal to one because one to any power would be equal to one. Proof is exactly the same as I was talking about a to the power of x being positive. Firstly, check it for natural numbers which is one times one times one times one. So for all natural x is obvious, that's one. Then negative natural numbers which is one over one which is one and then rational, et cetera. It's very, very trivial. So they're not going to address it anymore. Next property. All right, next property is actually related to these two graphs which I draw before. So basically it says that if a is greater than one, then for positive x, y would be greater than one and for negative x, y would be less than one. Again, remember the graph? Well, this is one. So this is y to the power of let's say, two to the power of x, something like this. Any number weighs greater than one. So it's increasing. This is level one. So if x is equal to zero, function is equal to one. Two to the power of zero as anything else to the power of zero would be one. Then function goes above one to the right and below one to the left. Now let's prove it. Forget about graph, let's prove it. Proof is exactly done in fashion as before. First, well, let's consider this case. So if x is natural, a to the power of n is a times a times a n times. Now each a is greater than one as we are saying. Now multiplying a greater than one number by greater than one number results in greater than one number. If it's not obvious, let me just prove it to you. Well, if a is greater than one, then it's just an interesting exercise. A can be represented as one plus d where d is greater than zero, right? So if a is greater than one, then d is some number greater than zero. Basically d is equal to a minus one, right? Now, therefore, a square is equal to one plus d square, which is one plus two d plus d square. Now, d is greater than one. So we add something to the one, which means we are getting a number greater than one. So multiplying a number greater than one by itself two times will result in number even greater than one. Well, basically this is the way how we can prove that any number of times we are multiplying number by itself, we will get the number greater than one. Well, the easiest way probably would be something like this. This is greater than one plus two g, right? Because there is one extra member here. So now let's go to a to the third degree. We will have one plus d to the third degree, which is equal to one plus d to the second degree times one plus d, which is greater than, I replace this with this, which is equal to one plus three d plus two d square, right? Which is greater than one plus three d. So look at this, a square greater than one plus two g, a cube greater than one plus three g. Now, it's very easy to prove, let's say by induction, that greater than one plus n times a. Basically that's enough to prove that it's greater than one. And the greater n is, the greater this number becomes. So let's just remember this particular inequality. I do encourage you to prove it by induction. Just check it for n is equal to one, which is kind of, which is, or n is equal to two, starting from n equals to two, which is obvious. And then if this inequality is true for some number, n is equal to k, try to go to k plus one. Do it yourself, it's very easy, and we will use it in some other place. But what it proves for us right now is that if a is greater than one, then for all natural numbers, which are integer positive, a to the power of n would be greater than one. We see it from here. How about negative natural numbers? What if we are talking about this? Well, since this is equal to this, this is greater, denominator is greater than one. So the inverse to a number which is greater than one is smaller than one. So we get this one. All right, so we have proved it for all integer numbers. How about rational? Quite frankly, it's exactly the same thing. Let's again start with this. If you have a to the power of p over q, which is by definition a q's root of a to the power of p, now this we have already proven that in this case, this is positive. This is greater than one, sorry. Positive obviously as well. Now, if you have a q's root from a number which is greater than one, you must have the result which is greater than one. Why? Otherwise, if it's less than one, and then you will raise it into the power of q, every time you multiply it by itself, it will be smaller and smaller and smaller because you assume that the number is less than one. So it cannot be greater than one in this particular case. So that's our best trick. Similarly, with negative rational numbers, since this is greater than one, then negative should be correspondingly inverse to a number which is greater than one, that is a smaller than one number. So I proved it for all rational numbers, and I told you I will not talk about irrational numbers, but let me just say it again, that irrational numbers are limit case of rational. So whatever the rational numbers in qualities are true for irrational numbers will always be true. Okay. Now, very similarly, what if a is less than one? Well, then I'm saying that for positive x, y would be less than one and for negative x, it would be greater than one. Well, I can prove it directly in exactly the same or similar fashion that I did it for a greater than one, but let me do it slightly differently. Look at this. a to the power of x is equal to one over a to the power of minus x. That's obvious, right? Because what is minus x? Minus x means one over one over a to the power of x, which is a to the power of x. Now, heading down that, I can basically refer everything to an opposite case because if a is greater, it is less than one and one over a is greater than one, right? So this to the power of x would be greater than one for x positive and would be less than one for x negative, right? So for a, it would be vice versa since this is equal to, so one, since one over a to the power of x is greater than one, then a to the power of x should be less than one for positive x. And if one over a over x is less than one for negative, then y over x should be greater than one for negative x. So that's basically kind of a easy logic which will reduce these cases to these cases. Just change the sign of the exponent. Okay, so back to our graph. Again, remember for a greater than one, it goes this way, for a less than one, it goes this way. Now, it looks like it's monotonically increasing for monotonically decreasing. So let's prove it, right? Why do I have such a nice and smooth curve? Let's prove the monotonicity. Is there such a word monotonicity? I don't know, maybe. So let's prove that the functions are monotonic for any a. For a greater than one, it's monotonically increasing for a less than one, it's monotonically decreasing. All right, so let's consider this case. Now, what is monotonically increasing? It means if we are increasing an argument, then the function is also increasing. That's what monotonic increase means. All right, fine. So let's just take two different arguments and we start with integer arguments, positive integer numbers. So let's say m less than n. So I will prove that if this is true, then a to the power of m would be less than a to the power of n. How can I prove that? Well, very easily, since m is less than n, then m is equal to n minus m to the power of n. Or if you wish, n is equal to m plus n minus m, same thing. So I will use this, actually. Now, a to the power of m, this one, therefore is equal to the power of n plus m n minus m. Equals. Now, you remember that whenever we are edging exponents, it means we can multiply exponential functions. Now, a is greater than one, right? So this, considering that m minus m is greater than zero, this is greater than one, right? It's base, which is greater than one, to the positive exponent, positive power. The result is, based on whatever we were just talking before that, greater than one. Since it's greater than one, then this number, a to the power of m, excuse me, is greater than a to the power of m, since this multiplier is greater than one. So we have proven that with increasing exponent, we increase the exponential function value as well, for greater than one base. So basically it's really looking like the curve goes up, but only for integer numbers. Now, how can we expand it to, let's say, rational numbers? Well, it's really very, very simple. Let's say we have two different rational numbers, p over q, which is less than r over s. I have to prove that a to the power of p over q is less than a to the power r over s. Now, how can I prove that? Well, let me first bring these two rational numbers to common denominator. Doesn't change the numbers themselves, so let's say I will use common denominator qs. So one would be p s over qs, less than r q over qs, right? So these are exactly the same. But now, oh, I'm sorry, qs. But now I have an advantage of having common denominator. So it's easier for me to consider the same kind of a theorem only for cases of common denominator because if denominator is not the same, I'll just bring them to common denominator. So let me forget about this particular case. I will use the case when denominator is the same. Let's say u over w, less than v over w. And I would like to prove from this that a to the power of u over w is less than a to the power v over w. So now I'm using the same w as denominator. Now, how can I prove this? You see, this means that u is less than v, right? They're all positive. We have already agreed that these are rational numbers with natural denominators and numerators and denominators. So obviously, since the denominator is the same, my numerators must be of the same relationship. Therefore, since I have already proven it for integer positive numbers for natural numbers, a to the power of u is less than a to the power of v for greater than one base a. Now, all I have to do now is extract the w's root from it, right? Because a to the power u over w is w's root from a to the power of u. And a to the power of v over w is again w's, the same w's root of a to the power of v. Now, if a to the power of u and a to the power of v are of this particular relation, then their w's roots also should be in exactly the same relation. Because if it's otherwise, then by multiplying greater number, the same number, the same w number of times, I will get greater number. So that's why the roots are supposed to be also in the same relation as a to the power of u and a to the power of v. Cannot be otherwise. If it's otherwise, I will just raise both in w's power and I will have a reverse equation here. So I've proved it for rational numbers as well. Fine. Now, these are all positive numbers. I was only talking about positive numbers and proved that. How can I prove it for negative numbers? Again, let's start from integers, negative integer numbers. Let's say I have minus m, well, let's first do this. This is minus m and this is minus n. So minus m is less than minus n. Now m and n are positive numbers. That's why I put minus in front of them. M and m and n are natural numbers. And since this one is to the left from this one, this is the relationship. I would like to prove that a to the power minus m is less than a to the power of minus n. Now, how can I prove that? Well, very easily. Now, if minus m less than minus n, then m, the positive corresponding positive, which is basically distance from zero, is greater than m. Therefore, from the previous case when both are integers, if this greater than this, then a to the m is greater than a to the m. Now, what is this? This is one over a to the m. And what is this? It's one over a to the power of m. Now, if denominators are in this relationship, then the whole fractions are in the opposite relation, right? Because if denominator is greater with the same numerator, then the fraction is smaller and vice versa. Okay, so basically that's what we have just proved, which means that the same monotonic property exists for negative integers as well. Now, absolutely the same way you go to the negative rational numbers and you have a mixture of negative and positive argument. It's even easier because we know that for greater than one base, everything to the right of the zero is greater than one and everything to the left of the zero is smaller than one, which means we still have exactly the same type of relationship between the values of the exponential function. Okay, now the case with a less than one, I will not consider at all because again it's quite obvious from the fact that if a is less than one, then one over a is greater than one. So all the properties of this case are true for this case just traversing the sign of the exponent. Whatever was true for positive exponent here will be true for negative here and vice versa. So that's why I'm not going to waste any time on this. That's all this. Next property. So monotone is to be proved now. Okay, we have to prove that a to the power of x for a greater than one increases to plus infinity if x goes to plus infinity. Okay, let me say it again. I'm using some symbolics here. So for this function, the function is already written here, so we don't have to think. If base is greater than one, if base is greater than one, therefore y goes up to plus infinity when x goes to plus infinity. So it's not only monotonically increasing, but it's increasing to infinity how to prove it. Very easy. We actually did something very useful. You remember I've proven this just before, couple of minutes before, for a greater than one, if you are raising it to the power of n, you will get very greater than this one. Now, how can I prove that this thing goes to infinity with increasing exponent n? Well, very easy. Let's choose any number n. And now I'm going to find such a number, lowercase n, that this thing exceeds this one. How? Well, let me choose the number n that this thing exceeds n. Now, when will this be greater than n? Well, obviously if n would be greater than n minus one divided by a, right? Minus one divided by a. So as long as I choose any integer number n greater than this number, then one plus n a would be greater than n and a to the n's power will be even greater. So it will exceed the number n. So for any number n, I can find lowercase n such that a to the power of lowercase n will exceed it, which means that for any level, I eventually will find the point where a to the power of n will exceed it. So that is actually the meaning of the statement that a to the power of x goes to infinity. At least we have proved this thing for all integers. So I can find an integer where it goes above certain level. But now let's think about the rational numbers. Since I have already proved that the function is monotonically increasing, once I found one particular point, let's say it's an integer lowercase n where we have exceeded this uppercase n, then everything above that would be even higher. So that's what actually is a proof that the function goes to plus infinity. It's monotonic and it can exceed any however large chosen number n. So what we have just proven is that for greater than one base, the function goes to plus infinity when the exponent increases to plus infinity. Now, how about the next my statement? Well, this is not a straight line, I'm sorry. Now my next statement is that when I go to the minus infinity when argument goes to negative infinity, then the function goes to zero. Well, this is actually quite obvious. Why? Because for negative number x for negative number x, I have this particular equation, right? This is a definition of negative exponent. So if a to the nth power, n is a natural number. So if a is a to the nth power goes to plus infinity, then one over a to the nth power goes to zero, right? One divided by a large denominator gives you a smaller fraction. And as soon as we understand that in this particular fraction, the denominator goes to positive infinity. We saw that the whole fraction goes to positive zero. Okay, fine. So I've got it for negative rational numbers, for negative integer numbers. So for negative rational numbers, it's obviously following from the fact that the function is monotonic. Okay, next. Okay, so we were actually comparing the values of the function based on different values of the exponent x. Now I would like to compare how the functions look if we are changing the base. This is not really an argument, but it would be interesting to find out how this function is different from this function. So my statement is that the bigger this particular base of exponential function for, let's talk about this case. So the bigger the base, the steeper the function goes, this is two to the power of x. And this is three to the power of x. So the bigger the base, the steeper it goes up to the right and faster it goes to zero to the left. Now with opposite a less than one, we will have a similar kind of a relationship that we will talk about this later. So right now, let me prove the following. That for a and b greater than one, if a greater than b, then, so if a greater than b, then a to the power of x is greater than b to the power of x for x positive and a to the power of x is less than b to the power of x is negative, right? So if this base is greater than this one, then for positive exponents, this graph is above this graph. But for negative exponents, this graph is below this graph. All right, how can I prove it? The same story is repeating again and again. First of all, let's prove it for natural exponent. So a to the n greater than b to the n. Natural are obviously greater than zero. How to prove that? Well, if a is greater than b, then whenever we multiply n times a, obviously each component of this multiplication would be greater than each component of that multiplication and that's why the result would be greater. Now, from this, we expand it to positive rational numbers because a to the power of p over q greater than b to the power of p over q because this is equal to q's root of h to the power of p and this is q's root of b to the power of p. If this is greater than this, then q's root should be also greater and that's what this is. All right, so we proved it for positive numbers. Now, how to prove, well, the opposite relationship to the negative numbers. Very easily, again, considering that for negative x, let me put a to the power of x for negative x, I will put it as minus absolute value of x and this is b to the minus negative absolute value of x. That's what negative actually exponent means. So, if x is equal to minus two, it's absolute value is two and that is minus two, same thing here. Now, this is equal to a to the power of x, this is equal to, sorry, this is equal to one over a to the power of absolute value of x. This is equal to b over to the power of x. Now, absolute value of x is a positive, so this denominator is greater than this denominator because we could just prove in it, but if we took inverse, we will have the inverse relation between them. So, if this denominator greater, then the fraction would be smaller. So, that's how we could prove in this one. Now, so basically what we have done here, we have proven that in these two cases, when both of them are greater than one, then the graph, both are increasing to infinity, in this case, and both decreasing to zero for x less than zero, but one function would be above another to the right of zero and would be below to the left of zero. Now, what if a and b are less than one? Well, the situation is absolutely symmetrical. Let me just write it down and I will just leave the proof to you. So, if a is less than one and b is less than one, let's say one half and one third. Well, basically the story should be exactly the same, but opposite. So, if this is the graph, this is let's say one half to the power of x. Now, if you have one third, then it should be one third to the power of x. Now, since my denominator is greater, then it goes down to zero faster. One half times one half times one half would be above one third times one third times one third. And on the left side, it would be exactly opposite. So, we can actually do the following. So, the function a to the power of x would be less than b to the power of x for x, yes, and a less than b. Okay, so if a less than b, then the value of a to the power of x would be smaller for negative and it would be bigger for positive. Let me check. The graph I know is correct. So, if a less than b, so a is one third and b is one half. So, one third, no, yes, yes, yes. a is one third, b is one half. No, it looks like I just mixed it. One third, one half, one third to the power of x one third to the power of x is smaller for positive x, I'm sorry, for positive x. This is for me. Right, it should be an opposite, right, okay. So, basically, this is exactly equivalent to the previous case and the proof is based on the fact that if a is less than one, then one over a is greater than one and one over b is greater than one. And that's why we can use the previous theorem for one over a and one over b. So, if a is less than one, then one over a is greater than one over b, right? And therefore, one over a in the power of x would be greater than one over b to the power of x for positive x. And that's what actually gives an opposite equation for denominators of these two members. And same thing with this one. All right, is that it? Yes, so, again, if base is greater than one, then the graph looks like this. This would be two to the power of x. This would be three to the power of x. The greater the base, the higher it is for the positive x and the lower it is for the negative. Now, if base is less than one, then situation is kind of an opposite. So, keep in mind these graphs. I think that all the properties which I was talking today are the foundation for these graphs. Now, I will explain in more details the story about graphs in the next lecture. So, this one was just an illustration, graphical illustration of all these properties. But again, the purpose of the properties is actually to draw nice graphs. Monetonicity greater than one or less than one based on the value of argument and the value of the base. So, all these properties are very important. And they're all based on the most important property of all exponential function, which is this. This. Okay, that's it for today. Again, next lecture will be about graphs. I do suggest you to take a look at the notes to this lecture which are accompanying the lecture on the Unisor.com website. And I also offer some exercises, problems and exams. For those who are registered, you're welcome to take exams as many times as you want. And I would also like to encourage the parents to basically be supervisors for your students. And the Unisor.com actually allows you to register as a supervisor and then you will be able to control the whole educational process of your students. That's it, thank you very much.