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so hello, dear friend, this is a new lesson on the channel, a young tutor, and today I would like to argue with you, let's do it so that you say that I don’t know logarithms and don’t know how to solve, but I will try to teach you and at the end of the lesson we will look at the result of our dispute and we will understand that I will be right, let 's start with the fact that in schools the logarithm is explained even well, but many guys still a little misunderstand does not understand the whole essence of logarithms do not understand the purpose of this algorithm, that is, some three terrible letters with which it is not I know they do not understand what to do, let's first figure out what it is and you will already understand that you got it all cool enough and about the dispute look if I teach you how to solve these problems at the end of the lesson, then I won and you will like it and subscribe to the channel if you cannot solve these problems at the end of the lesson, at least with me, then subscribe, leave and never watch me again, all let's start with what is the most important thing in the logarithm this is to understand what it is , let's write down, for example, some kind of logarithm is l k d logarithm a b is equal to c here, let's say for now these are incomprehensible letters and let's tell you one of them we will figure it out here a and b and c these are some numbers I hope you understand that you have probably seen some logarithm record there 2 of 8 is equal to 3 and so on and so on, the letter is called the base of the logarithm, you need to remember to take it as a fact, that is, how the logarithm is read in general, I can read, for example, this algorithm is how logarithm to base a from b here, respectively, and this is the base b of stage leg rhythmic expression, that is, b is some number, let's immediately fix with you what numbers a and b can be, that is, for c we do not have a certain definite interval c can be any the number of ogarev can be equal to anything but a and b are some definite numbers and these are positive numbers, that is, more than 0 and are not necessarily equal to one, we will now understand with you why these are necessarily positive numbers and now I will reformulate these three letters a little bit and tell you this is the logarithm of the number b to the base and this is such a number c and c is the degree of the base if I put my bases to this degree c I will get a head off, that is why do we even introduce the logarithm the logarithm we go to remove these extra words how, for example, in this case, what is it, what is a three, three is the degree in which I need to put this small number in order to get an eight, that is, 2 in the third power is 8 and by the definition of the logarithm, the logarithm base b is equal to c, then when the base in degree c is equal to the pawn, that is, we simply replaced the whole sentence with the logarithm, which is simply inconvenient to pronounce to speak, so that it is more convenient to consider the properties of logarithms, it is more convenient, but the property of these degrees was more convenient for some then the way to perform some actions with them just came up with these logs here and that's it for now on this we will stop, well, let's go over the properties then let's go before we go about the properties, let's decide something, for example, the logarithm of y to base 5 of 25, let's start with the easiest one again, I'll reformulate it burns to what extent I need to put this a small number five to get 25, you will tell me in the second, respectively, the answer for this logarithm will be two, let's also have a logarithm there from 3 to 27 to what degree do I need to put a three to get 27 you say well, so 3 by 3 is 9, that is, this is the second degree and again to 327, that is, degree 3 the whole definition of the logarithm, you and I looked, we figured it out and drove on all the logarithm have some properties, let's write down all these properties, especially the most important ones, we'll talk about them, and unfortunately you will have to either to understand them and somehow learn without you learning, but just at the level of understanding, or learn them, that is, the simplest I read here is to learn these properties and calm to use, but to understand the main thing where these properties are applicable and to not applicable, let's start with the first one, this property, which, in principle, when solving problems for solving equations, is infrequent, but I use it, we will need to look at it, see if I have a number and it will melt under the degree, the degree which will be the logarithm to the base and from the number b, then this number is well, then the answer to this whole expression will be just hesitate, let's think of, for example, there is 3 to the power of the logarithm of 3 from there, let's go well 27 to let's say by the property the answer will be the number 27 let's check the logarithm base 3 of 27 what is it this is the degree in which you need to substitute a three to get 27 convicts we just solved to the third degree gross excellent this means that 3 in the third degree equals 27 here is our 27 here is where it came from the first property that you just need to visually at least remember and use it is the main thing that the base of the logarithm and the base is this were the same numbers the second property if I meet just such a record, I can immediately say that this is one why let's check the logarithm there 3 of 3 which will be equal so to what degree do I need to substitute this triple to get a triple you cannot say but in the first degree we put the number will not change, respectively, the answer is ones, so we can write absolutely any number of the river logarithm there 144 5 8 of the same number will be equal to one because I need to put in the first degree so that it does not change the third property of the logarithm and in base and from one it will be equal let's figure it out, for example, the logarithm of 3 is the logarithm base 3 of one which will be equal to and so to what degree I need to substitute a three to get one if we remember the properties of the degrees we had one very cool property there that the number a standing in the zero degree is any number a standing of the zero degree is always equal to one, that is, I need to put three in the zero degree to get to be one, respectively, the answer for a given logarithm will be zero, this applies again to absolutely any number, because if I substitute any number for a zero degree, I will get one, the most important thing is that our reasons should not be negative and should not be equal to one, so the fourth property well, yes 4 this property is already used quite often, but there is also in the exam in the second part and at school you probably had it or will, so let's see if we have the sum of two logarithms with the same base there and we already have this sum of logarithms we can replace as the logarithm, therefore, the base a from the product of b is the nation, that is, for example, let's do something, for example, I have the logarithm of the logarithm of 2 of 4 plus the logarithm of 2 of 8, according to my property it will be equal to the logarithm of 2 of 4 multiplied by 832, let's check the logarithm of 2 logarithm base 2 of 4 what is it this is the second degree, that is, two here 3 degree I will add up I get that the five let's check the five is it in what degree you need to put a two that will get 32 ​​2 by 2 4 4 by 2 8 8 by 2 1616 by 230 25 degree so we got it and the fifth property is quite similar and we will not even paint it, I think it is identical in the fourth only with a sign minus and equal to ours, just about the difference, our difference will be equal to the logarithm and at bay divided by tions, it's easy I think that yes, in powers, let's see, let's see a new one, let's open it, we get the sixth property, yes there will be the sixth property if I have a logarithm base out b & b for example, it costs some degree c what can I do with it I can throw out the degree c for the logarithm by multiplying by my logarithm, that is, multiply c by the logarithm and along the arc and let's imagine everything, for example, the logarithm 2 leaving 8 in the third degree will be pretty hard to count this is why we just throw out the triple and find just this expression logarithm 2 of 8 new to base 2 of 8 this triple 3 by 3 and 9 we could prove it in principle we could imagine how we could to represent the eight already as two in the third degree and all those standing in the third degree are multiplied would get the logarithm of 2 of 2 in the ninth degree again we have a nine, this would go here but here we would get 9 multiplied by the logarithm of 2 of 2, that is, one 9 by 29 per one, nine is the sixth yes, guys, I don't understand that taking 7 further there will be a sim of the seventh property, we figured out what we disassembled blew up the degree of wpad logarithmic and now we substitute this degree of porridge if I have and my bases are in degree c what can I do I can throw out this degree again for the logarithm, but just not just throw it out, but throw it out exactly in the denominator as it will look like it will be equal to 1 divided by c multiplied by the logarithm of a from b, that is, let's check well, let's sign something to make it more clearly there, the logarithm of 5 standing in the third degree of 25 will be equal to three, we throw out the denominator, we get one third multiplied by the logarithm of 5 of 25, that is, two, that is, two thirds, here mo th answer I think that so far everything is easy and everything is clear and let's just write such a thing, this property is, of course, very rare, but you still need to know and understand this property is called a transition to a new base, see if I have a logarithm, but here I am I can replace this logarithm in what way if I suddenly somehow need to represent my logarithm for something, but not on the basis of x on the basis of some price, I can write down the logarithm as well as the logarithm on the base c from, but from rhythmic friend divided by the logarithm of a base b from a, that is, we kind of start with a large number, go to a small one, let's check well, let's write down something, for example, we have a logarithm of 2 of 8 and let's say I want to bring it all to a base of some kind 3 that is, but here it will probably not be a very beautiful chest, well, let's try that is, the logarithm of 3 and in the numerator of the logarithm of 3 in the denominator, we are in base 3 here eight, here two is all that is, but we have numbers like ral such that they will not give us any good answer, but I changed the basis, I can already further there something with something to reduce, let's say if I have some further actions and so on, etc., so I think that the property of logarithms is over, yes we examined the degrees we examined the change of the base, I think everyone see what kind of problems there are in general there are a lot of problems in pots, and when we have something like this, for example, the base 5 logarithm from there xa, but I give an example, they will not completely write down the expression will be equal to the logarithm base 5 of 8, since I have both here and here the logarithm I can just equate my friends rhythmic why can I do this, let's think what will be the answer for this, that is, some degree to which I will climb five that get x for this, some degree to which I will raise a five to get 8, but if these degrees are equal, it means that I raise a five to the same degree, and here it is already eu It’s my eight of correspondence that was erected just says that x is 8 and that's it, let's solve the problems that we planned to solve at the very beginning, these are the problems because of the open bank, the game is the problems of the first part, let's solve these equations, that's what I was talking about Regarding Lugarev about the equality of the logarithms, the most important thing is that the base is the same, so I just equated x minus 2 to 11 well, and I transfer the two to the right side to get that 11 plus ba 13 here 15 minus x equals seven, since again the same base is in the right side well let's move x to the right and the seven to the left, that is, we get 15 minus 7 in the left, then that the eight is equal to the ax, that is, x is 8, with this I think that this already needs to be done according to the classical definition of the logarithm to what degree I need to raise 3 to get here it is yours they are in the fourth, that is, 3 in the fourth power is 8x minus 15 right, let's count 3 in 4 3 in 3 99 in 3 2727 for 3 of them, probably 81 like 81 that is 81 equals 8x minus 15 we transfer 15 to the left we get 8 x equals 8181 plus 15 to 96 then x equals 96 divided by 8 it seems to be 12 so something like this let's decide again to what degree we need to build a four to get 4x minus 8 in the third the degree, respectively, 4 in the third degree is 4x minus eight it remains just to find 4 in the third degree 4 by 4 1616 by 4 probably 64 64 is equal to 4x minus 8 the eight is transferred to the left side that we get 64 plus 8 is equal to 4x so equal to 4x then here I have that 72 is equal to 4 xx will be what is equal there, then if we give one here 32 is 818 and let's work with such expressions, let's see what to do with them I'll hang 3 to the power of the logarithm of 9 from 4x plus 1 so well, this is equal to 9 until there are in fact, two options, the first option, we can use this property here, our first, but we definitely need to somehow get the same base of the logarithm and the same number, this is what can be done here, for example, we can say that nine is what it is no, probably it will not be the best not the best no no we will get one second here no let's do it a little differently look here what we did here we can get something similar here if I present now the logarithm of 9 from 4x plus 1 and nine I will present it as 3 in the second degree that I will get that I have three in some degree equal to 3 in the second degree what conclusion can I make that the logarithm of 9 from 4x plus 1 is equal to two, that is, we raise the nine to two and get 4 x plus one here we are 81 equals 4x plus 14 x equals 81 minus 1, that is, 80 x equals 84 twenty all and give the last 2 logarithms 4 2 x plus 2 equals 4 again 4 I immediately see that it is 2 in the second power, that is, 2 in the power of logarithm 4 here 2 x plus 2 is equal to 2 in the second degree again here I have two raised to some extent equal to 2 raised to the second power to what conclusion can I make that these degrees are equal I will not be equal only when I have these degrees are equal to each other Responsibly, I get the logarithm of 4 2 x plus 2 equals oh, our rules say twos again 4 I raise to the second power to get 2 x plus 24 in the second 16 equals 2x + 22 x equals 14 and it seems that x equals 7 x equals 7 as- then today we analyzed all the properties of the logarithm, we looked at how they work and we decided due to the knowledge of these properties and there are a couple of additional new knowledge that, in principle, is a bit similar to these properties, about equality, yes, we solved several tasks from the first part and the cat profile level, of course, that's why you just have to like it, subscribe to the channel, learn the properties of logarithms and become an excellent student, let's all for now.