Логарифмы с нуля за 20 МИНУТ! Introduction to logarithms. — Transcript

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1. 0:00so hello, dear friend, this is a new lesson on the channel, a young tutor, and
2. 0:04today I would like to argue with you, let's do it so that you say
3. 0:11that I don’t know logarithms and don’t know how to solve, but I will try to teach you and
4. 0:18at the end of the lesson we will look at the result of our dispute and we will understand that I will be right, let
5. 0:35's start with the fact that in schools the logarithm is explained even well, but many
6. 0:49guys still a little misunderstand does not understand the whole essence of logarithms do not understand the
7. 0:56purpose of this algorithm, that is, some three terrible letters with which it is not I know they do not
8. 1:03understand what to do, let's first figure out what it is and you will already understand that you
9. 1:09got it all cool enough and about the dispute look if I teach you how to solve these problems at the end of the lesson,
10. 1:19then I won and you will like it and subscribe to the channel if you cannot
11. 1:26solve these problems at the end of the lesson, at least with me, then subscribe, leave and never
12. 1:33watch me again, all let's start with what is the most important thing in the logarithm this is to understand what it is
13. 1:41, 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
14. 1:53for 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
15. 2:01numbers I hope you understand that you have probably seen some logarithm record there 2
16. 2:07of 8 is equal to 3 and so on and so on, the letter is called the base of the logarithm,
17. 2:18you need to remember to take it as a fact, that is, how the logarithm is read in general, I
18. 2:27can read, for example, this algorithm is how logarithm to base a from b
19. 2:33here, respectively, and this is the base b of stage leg rhythmic expression, that is, b is some number,
20. 2:43let's immediately fix with you what numbers a and b can be, that is, for c we do not have a certain
21. 2:51definite interval c can be any the number of ogarev can be equal to anything but a
22. 2:57and b are some definite numbers and these are positive numbers, that is, more than 0 and are not necessarily
23. 3:06equal to one, we will now understand with you why these are necessarily positive numbers and
24. 3:14now I will reformulate these three letters a little bit and tell you this is the logarithm of the number b to the
25. 3:24base and this is such a number c and c is the degree of the base if I put my bases
26. 3:34to this degree c I will get a head off, that is why do we even introduce the logarithm the logarithm we go to remove
27. 3:42these extra words how, for example, in this case, what is it, what is a three,
28. 3:48three is the degree in which I need to put this small number in order to get an
29. 3:55eight, that is, 2 in the third power is 8 and by the definition of the logarithm, the logarithm base
30. 4:01b 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,
31. 4:11which is simply inconvenient to pronounce to speak, so that it is more convenient to consider the properties of
32. 4:17logarithms, it is more convenient, but the property of these degrees was more convenient for some then the way to
33. 4:23perform some actions with them just came up with these logs here and that's it for now on this
34. 4:28we will stop, well, let's go over the properties then let's go before we go about the properties, let's
35. 4:37decide something, for example, the logarithm of y to base 5 of 25, let's start with the easiest one
36. 4:46again, I'll reformulate it burns to what extent I need to put this a small number five
37. 4:53to get 25, you will tell me in the second, respectively, the answer for this logarithm
38. 4:59will be two, let's also have a logarithm there from 3 to 27 to what degree do I need to put a three
39. 5:07to get 27 you say well, so 3 by 3 is 9, that is, this is the second degree and again to 327,
40. 5:13that is, degree 3 the whole definition of the logarithm, you and I looked, we figured it out and drove
41. 5:20on all the logarithm have some properties, let's write down all these properties, especially the most
42. 5:28important ones, we'll talk about them, and unfortunately you will have to either to understand them and somehow
43. 5:36learn without you learning, but just at the level of understanding, or learn them, that is, the
44. 5:43simplest I read here is to learn these properties and calm to use, but to understand the main thing where
45. 5:49these properties are applicable and to not applicable, let's start with the first one, this property, which, in principle,
46. 5:56when solving problems for solving equations, is infrequent, but I use it, we will need to look at it,
47. 6:05see if I have a number and it will melt under the degree, the degree which will be the logarithm
48. 6:16to the base and from the number b, then this number is well, then the answer to this whole expression will be
49. 6:22just 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
50. 6:31to let's say by the property the answer will be the number 27 let's check the logarithm base 3
51. 6:41of 27 what is it this is the degree in which you need to substitute a three to get 27 convicts we
52. 6:48just solved to the third degree gross excellent this means that 3 in the third degree
53. 6:52equals 27 here is our 27 here is where it came from the first property that you just need
54. 7:00to visually at least remember and use it is the main thing that the base of the logarithm and the base is
55. 7:07this were the same numbers the second property if I meet just such a record, I can immediately say
56. 7:18that this is one why let's check the logarithm there 3 of 3 which will be equal so to what
57. 7:30degree do I need to substitute this triple to get a triple you cannot say but in the first degree
58. 7:35we put the number will not change, respectively, the answer is ones, so we can write absolutely any
59. 7:41number of the river logarithm there 144 5 8 of the same number will be equal to one because I need to
60. 7:51put in the first degree so that it does not change the third property of the logarithm and in base and
61. 8:01from one it will be equal let's figure it out, for example, the logarithm of 3 is the logarithm base 3 of one
62. 8:12which will be equal to and so to what degree I need to substitute a three to get
63. 8:19one if we remember the properties of the degrees we had one very cool property there that the
64. 8:27number a standing in the zero degree is any number a standing of the zero degree is always equal to one,
65. 8:32that is, I need to put three in the zero degree to get to be one, respectively, the
66. 8:38answer for a given logarithm will be zero, this applies again to absolutely any number,
67. 8:45because if I substitute any number for a zero degree, I will get one, the most
68. 8:54important thing is that our reasons should not be negative and should not be equal to
69. 8:59one, so the fourth property well, yes 4 this property is already used quite often, but
70. 9:12there is also in the exam in the second part and at school you probably had it or will, so
71. 9:17let's see if we have the sum of two logarithms with the same base there
72. 9:26and we already have this sum of logarithms we can replace as the logarithm, therefore, the base a from the product of
73. 9:31b is the nation, that is, for example, let's do something, for example, I have the logarithm of the logarithm of 2 of
74. 9:434 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,
75. 9:55let's check the logarithm of 2 logarithm base 2 of 4 what is it this is the second degree, that is,
76. 10:02two here 3 degree I will add up I get that the five let's check the five is it in what degree
77. 10:10you 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
78. 10:19got it and the fifth property is quite similar and we will not even paint it, I think it is
79. 10:26identical in the fourth only with a sign minus and equal to ours, just about the difference, our difference will be
80. 10:37equal 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
81. 10:47new one, let's open it, we get the sixth property, yes there will be the sixth property
82. 10:55if I have a logarithm base out b & b for example, it costs some degree c what can I do with it
83. 11:08I can throw out the degree c for the logarithm by multiplying by my logarithm, that is, multiply c by the logarithm and
84. 11:16along the arc and let's imagine everything, for example, the logarithm 2 leaving 8 in the third degree will be pretty hard to
85. 11:28count this is why we just throw out the triple and find just this expression logarithm
86. 11:372 of 8 new to base 2 of 8 this triple 3 by 3 and 9 we could prove it in principle
87. 11:44we could imagine how we could to represent the eight already as two in the third degree and all those
88. 11:51standing in the third degree are multiplied would get the logarithm of 2 of 2 in the ninth degree
89. 11:59again we have a nine, this would go here but here we would get 9 multiplied by the logarithm of 2 of 2, that is,
90. 12:10one 9 by 29 per one, nine is the sixth yes, guys, I don't understand that taking 7 further there
91. 12:20will be a sim of the seventh property, we figured out what we disassembled blew up the degree of wpad logarithmic
92. 12:28and now we substitute this degree of porridge if I have and my bases are in degree c what can I do
93. 12:38I can throw out this degree again for the logarithm, but just not just throw it out, but throw it out exactly
94. 12:44in the denominator as it will look like it will be equal to 1 divided by c multiplied by the logarithm of
95. 12:51a from b, that is, let's check well, let's sign something to make it more clearly there, the
96. 12:57logarithm of 5 standing in the third degree of 25 will be equal to three, we throw out the denominator, we get one
97. 13:07third multiplied by the logarithm of 5 of 25, that is, two, that is, two thirds, here mo th answer I think
98. 13:16that so far everything is easy and everything is clear and let's just write such a thing, this property is, of course,
99. 13:26very rare, but you still need to know and understand this property is called a transition
100. 13:33to a new base, see if I have a logarithm, but here I am I can replace this logarithm in
101. 13:40what way if I suddenly somehow need to represent my logarithm for something,
102. 13:47but not on the basis of x on the basis of some price, I can write down the logarithm as well as the logarithm
103. 13:54on the base c from, but from rhythmic friend divided by the logarithm of a base b from a, that
104. 14:04is, we kind of start with a large number, go to a small one, let's check well, let's write down something,
105. 14:10for 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
106. 14:183 that is, but here it will probably not be a very beautiful chest, well, let's try that is, the logarithm of
107. 14:243 and in the numerator of the logarithm of 3 in the denominator, we are in base 3 here eight, here two is all
108. 14:32that is, but we have numbers like ral such that they will not give us any good answer, but I changed the
109. 14:38basis, I can already further there something with something to reduce, let's say if I have some
110. 14:43further actions and so on, etc., so I think that the property of logarithms is over,
111. 14:54yes we examined the degrees we examined the change of the base, I think everyone see what kind of problems
112. 15:05there are in general there are a lot of problems in pots, and when we have something like this,
113. 15:13for example, the base 5 logarithm from there xa, but I give an example, they will not completely write down
114. 15:22the expression will be equal to the logarithm base 5 of 8, since I have both here and here the logarithm I can
115. 15:32just equate my friends rhythmic why can I do this, let's think what will
116. 15:38be the answer for this, that is, some degree to which I will climb five that get x
117. 15:44for this, some degree to which I will raise a five to get 8, but if these degrees are equal,
118. 15:50it means that I raise a five to the same degree, and here it is already eu
119. 15:55It’s my eight of correspondence that was erected just says that x is 8 and that's it, let's
120. 16:02solve the problems that we planned to solve at the very beginning, these are the problems because of the open bank, the
121. 16:11game is the problems of the first part, let's solve these equations, that's what I was talking about Regarding
122. 16:17Lugarev about the equality of the logarithms, the most important thing is that the base is the same, so I just
123. 16:23equated x minus 2 to 11 well, and I transfer the two to the right side to get that 11 plus ba 13 here 15
124. 16:34minus x equals seven, since again the same base is in the right side well let's move x
125. 16:42to 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,
126. 16:49that is, x is 8, with this I think that this already needs to be done according to the classical definition of the
127. 16:54logarithm to what degree I need to raise 3 to get here it is yours they are in the fourth, that
128. 17:02is, 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
129. 17:14them, probably 81 like 81 that is 81 equals 8x minus 15 we transfer 15 to the left we get 8 x equals
130. 17:268181 plus 15 to 96 then x equals 96 divided by 8 it seems to be 12 so something like this let's
131. 17:38decide again to what degree we need to build a four to get 4x minus 8 in the third the degree,
132. 17:45respectively, 4 in the third degree is 4x minus eight it remains just to find 4 in the third degree
133. 17:514 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
134. 18:0264 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
135. 18:22if we give one here 32 is 818 and let's work with such expressions, let's see what
136. 18:31to 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
137. 18:45are in fact, two options, the first option, we can use this property here, our first,
138. 18:51but we definitely need to somehow get the same base of the logarithm and the same number,
139. 18:57this is what can be done here, for example, we can say that nine is what it is no, probably it
140. 19:07will not be the best not the best no no we will get one second here no let's do it a little
141. 19:14differently look here what we did here we can get something similar here if
142. 19:23I present now the logarithm of 9 from 4x plus 1 and nine I will present it as 3 in the second degree that
143. 19:32I will get that I have three in some degree equal to 3 in the second degree what conclusion can I
144. 19:37make that the logarithm of 9 from 4x plus 1 is equal to two, that is, we raise the nine to two and get 4
145. 19:47x plus one here we are 81 equals 4x plus 14 x equals 81 minus 1, that is, 80 x equals 84 twenty all and
146. 20:01give the last 2 logarithms 4 2 x plus 2 equals 4 again 4 I immediately see that it is 2 in the second power,
147. 20:13that 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
148. 20:21raised to some extent equal to 2 raised to the second power to what conclusion can I make that
149. 20:27these degrees are equal I will not be equal only when I have these degrees are equal to each
150. 20:32other Responsibly, I get the logarithm of 4 2 x plus 2 equals oh, our rules say twos again 4 I
151. 20:43raise to the second power to get 2 x plus 24 in the second 16 equals 2x + 22 x equals 14 and it
152. 20:55seems that x equals 7 x equals 7 as- then today we analyzed all the properties of the logarithm, we looked at how
153. 21:08they work and we decided due to the knowledge of these properties and there are a couple of additional new knowledge
154. 21:15that, in principle, is a bit similar to these properties, about equality, yes, we solved several
155. 21:23tasks from the first part and the cat profile level, of course, that's why you just have to
156. 21:31like it, subscribe to the channel, learn the properties of logarithms and become an excellent student, let's all for now.