 Hi, I'm Zor. Welcome to Unisor Education. I would like to talk about logarithms. It's kind of natural sequence after we were talking about exponential functions. All right. This is an introductory lecture, and I'll talk about like definition of logarithm, and it's very, very basic properties without any details and no difficult stuff at all. All right. So let's start with exponential function. I would like to remind the couple of things about exponential function. Number one is the function is defined for positive a and not equal to 1. Because 1 to any power will probably result in 1, so it's not really an interesting function. So we are talking about a function of this type. Now, it's obviously defined for all real values of x and y is also real and it's greater than zero. I'm just reminding basic properties of this. Another basic property is that the function is monotonic. It's monotonically increasing with a greater than 1 and it's monotonically decreasing for values of a less than 1 and greater than zero, of course. Now, if you refer back to the lecture about monotonic functions, I was actually talking about monotonically increasing functions, but it doesn't really matter. It's equally applicable to monotonically decreasing as well. You know that monotonic function establishes the one-to-one correspondence between domain and the range. And in particular, it follows from this monotonicity of the function and the fact that it's one-to-one correspondence. There is an inverse function, so there is always inverse function in this case. Well, good. So there is an inverse function in this particular case. And if the exponential function establishes the relationship between all real numbers, one-to-one correspondence actually between all real numbers and positive real numbers, then inverse function establishes the one-to-one correspondence between positive real numbers and all real numbers. Well, actually this function is called the logarithmic function. So if you know how to find from a real number x, a to the power of x, which is equal to y, and we know that there is an inverse function, it means that if y is given to you, you can always find such an x that a to the power of x would be equal to y. Well, now, since we're talking about reverse function, and usually x is used as an argument and y as a function, so we can talk about function, which is the following. It's symbolically written as this, and what it means actually. It means that you have to find such a variable of the exponent, which is y. If a is raised to this power, you will get x. So if a is raised to the power logarithmic x base a, it's x. This is a definition, definitely sure. It's not a theory, it's a definition. The logarithm of some value x, and it should be positive since we're talking about the range of this function. So logarithm of the positive value x base a is a value which is used as an exponent with the base a will give x. So since this is a definition, there is nothing to prove about this, but what we can say is that it also, this function also establishes the one-to-one correspondence, in this case between all the values where the result of this function can be, which is positive real values. It's one-to-one correspondence with all real values, which can be the value of the logarithm. So this is logarithm x by the way of a. From all positive real values establish one-to-one correspondence and monotonic by the way function to the all real values. All right, so that's basically the definition, not too much to it. Now let's talk about monotonicity. Now we know that the monotonic function, this monotonic function, is increasing when a is greater than zero. Now if the function increases, then inverse function is also monotonically increases, right? And it's obviously, I actually, I can prove it actually, it's very easy to prove. Monotonic function is if u less than v, then f of u less than f of v. Now the function which works backwards, let's call it g function, let's use, so basically g function transforms f of u into u and it transforms f of v into v, right? Now, so if this is less than this and it's given, then this might be less than this, because if it's not, then we can apply the function again, f, and back to these values, and it will be an opposite relationship. So if this is less than this, this cannot be greater or equal or anything like that, because then the function would not be in the same relationship as we were actually talking about. So this is elementary stuff. Now, since we have established that logarithmic function establish the monotonic one-to-one correspondence between all real, positive real numbers and all real numbers, it's monotonically increasing whenever the exponential function is increasing. So you know that exponential function is increasing when a is greater than one and decreasing when a is less than one. So similarly, inverse function is increasing when a is greater than one and decreasing when a is less than one. Now, I think it's time to exemplify whatever I was talking about, and I have a couple of examples. What is logarithm base 2 of 256? Now, what did we know about logarithm? Logarithm of a number is a power which is, if used with this base, will give this number. So what the power I should raise the 2 to get 256? Well, 8. 2 to the power of 8 is 256. That's why this is equal to 8, because 2 to the power of 8 gives you 256. It's directly from definition. I didn't really calculate anything. I basically guessed, yes, I guessed the value of this function. I guessed that if I would take 8, then knowing that 2 to the power of 8 is 256, this is the right value. And there is no other value, because it's one-to-one correspondence. There is only one value. It's monotonic function. Now, next example, logarithm base 10 of 0.01. Okay, what is it? Again, what power should I use with the base 10 to get 100s? Well, remember that 100s, this is 10 to the power of minus 2, right? That's the definition of negative 2 as a power. So that's what I have to do this. Again, I kind of guessed that if 10 is raised to the power minus 2, I get 0.01. That's why minus 2 is the value of this logarithm. And the third example I have is logarithm of 1 base 2.5. I don't want to scare you. I put base 2.5. It's kind of not exactly the nice number, but I put 1 here. Now, let's just remember that any number to the power of 0 will give you 1. Remember this, right? So that's why this is equal to 0, because 2.5 to the power of 0 will give 1. So always, whenever you see something like this, it signifies the power into which I have to raise the base to get the argument. Okay? Now, as a couple of particular cases, special cases with logarithms, there are certain bases which are more preferable than others. Primarily, base 10 is very often used in different calculations. So whenever we're talking about function with base 10 as a logarithm, there is a special notation about this function. It's either no base at all, and 10 basically is assumed. Sometimes, in some, I think it's non-English literature, they use just lg as this particular type of logarithm. And it's called common or decimal logarithm. Now, another type of logarithm. There is a number which is called E in mathematics, which is approximately 2.721. It's actually a irrational number, and it's very important in mathematical analysis for different purposes. I'm not talking about the purposes right now, I'm talking about the logarithm which has the base E. It has a very, very important function in analysis, and that's why there is a special notation. The logarithm is with the base E is called natural logarithm, and its notation is ln. And it is natural, believe me, there are many different reasons why this particular logarithm is called natural. All right, so these are just the terminology. Now, I would like to present a very important property of logarithm, which is probably the most important property. And I would actually be willing to say that this property is the reason why people kind of invented logarithms and use the logarithms. And here is the property. Now, you remember this property of exponential function. Now, if you don't, I refer you to previous lectures about exponential functions. This is a typical kind of a thing. Just for example, 3 to 15 is equal to 335. No, sorry. So, exponents are added together whenever you multiply the exponential expressions. Now, let's call this u, this v, actually, let's call this u, this v, and this w. So, what can I say? Well, w is equal to a to the power of p plus q. Now, what does it mean? p plus q is an exponent, the power actually, I can raise a to get w. What is this? This is, by definition of logarithm, the power I should raise a to get w, which is the same thing. So, it's p plus q. So, from this, I can write this. They mean exactly the same thing. This is a definition of logarithm, basically. If you have this, p plus q is the power I have to raise a to get w. p plus q is, by definition, logarithm is a power I have to raise the base to get the argument. That's the same thing. Similarly, p is logarithm u at base a. Why? Because if I raise a to the power p, I will get u. That's what it is. Again, this is just the definition of logarithm. I didn't do any logical derivation here, just a specific definition. Similarly, q is logarithm v, the power of a. So, what do we see here? Well, p, q, p plus q, right? So, using the right side of this equation, I would say that logarithm u plus logarithm v with the same base, this plus this is equal to this, which is logarithm power base a of w. But w is u times v, right? From this, w is u times v. Here is, basically, the formula. In logarithms, this formula is equivalent to this one in exponential functions. The logarithm of the product is equal to sum of logarithms. It's using the same base everywhere, obviously. Same thing as this. The exponent of a sum is equal to product of exponents. It's, in some way, it's similar. But this is the function. This is inverse function, and that's why the property actually is inverse. From summation we go to product, from product we go to summation. Kind of symmetrical, right? Now, my claim is that this property is so important that it actually is the reason why logarithms exist. Now, here is why. Couple of days ago, it's just a personal example, by the way. Couple of days ago, I wanted to calculate some probability. And it was something like, what's the probability of, let's say, 80 people out of 100 having some property? So, in the calculation of this probability, I had to calculate something like this. Now, this is factorial. It means 1 times 2 times 3, et cetera, times 100. This is the product of all numbers from 1 to 80, and this is the product of all numbers from 1 to 20. How can I possibly calculate this? There is absolutely no way. I can't even program it in the computer, because it will go outside of the range of the computer program, outside of the computer memory or something. However, I can use this particular function. By the way, similar to this, I can say that logarithm of u divided by v is equal to logarithm u minus logarithm v. I will basically prove this. It's very easy to prove. But in any case, what I will do, instead of calculating this, I will calculate logarithm of this. Logarithm of this result of this division is equal to logarithm of 100 factorial minus minus logarithm 80 minus logarithm 20 factorial. Now, this is 1 times 2 times 3, et cetera, times 100. So, instead of doing this, instead of multiplying 1 by 2 by 3, et cetera, I will basically do summation. It's logarithm 1 plus logarithm 2, et cetera, plus logarithm 100, right? Because the logarithm of a product is equal to sum of logarithms. Similarly, this also, this is a sum. This is a sum, and this is a sum. Now, once I did this, I can calculate it. I can put it into any program, and the program will calculate without any problems. And then, after I have figured out what my logarithm of this is, all I have to do is I have to raise 10 into this power to get the value of this. So, without logarithms, I wouldn't even be able to do this calculation even with the computers, because the numbers are too large. With logarithms, by converting whatever the very complicated expression you have into its logarithms, and calculating in the logarithm, and then exponentially transfer it back, that's how the calculations are made. Basically, again, this is, in my personal view, this property is the justification for existence of the logarithms. So, what's the main, kind of, let me sum up whatever I was talking today about. To sum it up, I can say that number one, you have to understand the definition of the logarithms. And the definition, basically, is in this formula. A to the power log x base A is equal to x. This is a definition. What power should I use for a base A to get x? That's the logarithm of x by the base A. That's number one. And number two, another very important property is this. The logarithm of product is equal to sum of logarithms. These are two main properties of the logarithm which you really should remember, and keep it always in mind. Everything else can be derived from this. Just as an example, from this, we can very easily derive this. This is the property. If under the logarithm, under the argument you have some kind of a power of something, then the power can be brought outside of the logarithm and whatever remains the base of this exponential function would be an argument of the logarithm. Now, if you have, for instance, a natural exponent here, let's say u to the nth. This is u times u, et cetera, times u. So if I will take logarithm of this, it will be equal to logarithm of this and this is the logarithm of a product, which is sum of logarithms. It's all base A, and this is n times, because the product A to the power of n is the product of n times product of u by itself, which is exactly equal to n times log u by the base A. So as you see, the exponent is brought outside of the logarithm. But this is for a natural n, because then we can write u to the power of n as u times u times u. Now, what about any power? Power, as you know, an exponential function can be any real number. And this particular property is still true, and here is why. Now, let's go from the definition. As I was saying, the definition and elementary properties is enough. What is this? This is, let's call it w is equal to log u to the power of delta, I'm saying. What does it mean? It means A to the power of w equals to u to the delta, right? That's what it means. A to this power gives you an argument. Well, if I'm claiming that this is equal to this, well, let's check. We have to check that A raised to this power should equal to this. If I will prove this, then I will prove this, because this is exactly the same thing. If I would like to show that this is logarithm of u to the power of delta by the base A, I have to prove that A to this power gives me u to the delta. Now, is it true or not? Well, again, you remember the property of exponential functions. I hope you remember this. If you don't, again, go back to exponential functions lecture. A to the product of two numbers is A to the first number and the result to the second. Or they can be reversed, actually. It doesn't really matter because the product is commutative. So sequential raising into power basically results in the multiplication of the exponents. So here, I'm using this property. Instead of delta would be p. Instead of log u, log u by the base A, it would be q. So I will use this thing. So this is my pq. q is this. So it's A to the power of log u base A to the power of delta, delta is p. What is this? What is log u by the base A if A I raise into this power? This is a definition of logarithm of log u. Because this is the power I should raise A to get u. So let's raise A to this power. I will get u. So this will be u still to the power of delta. So that's a very simple proof, actually, which is based only on the definition of logarithm and this property of exponential function. Basically, that's how I would like to finish this introduction. This is just an introduction because the properties of logarithms are much more numerous and there are some interesting problems which I will solve with you. Thanks very much for listening to this lecture. I do encourage you to go into Unisor.com and basically take the whole course. This is just one small part of it. And it would be great if you can register and take exams. That's very, very useful. So I do recommend it to you. Thank you very much.