Introduction to logarithm properties | Logarithms | Algebra II | Khan Academy

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Welcome to the algorithm properties lesson There will be many practical exercises And if one of these characteristics does not convince you And you want to prove it, I did three or four shows To install these properties But what I would like to do is show you the characteristics and then explain how to use them This requires more practical exercises Now let’s do a quick review of the algorithms If you assume a – oh this is not true Let’s see to see I want to change this – let’s start Suppose, for example, that a – let me start – a ^ b = c So – aa b = c We can express this relationship in another way instead of Writing an exponent, we’ll write it as a log So we can say that the logarithm of the basis is a for c = b Both express the same thing, and what differs from it is the result In the first case, you know the value of both a and b and from them you will find the value of c This is the basics benefit for you In the second case, you know the value of a where when Raise it to a power you will get c From here you can find the value of b So all this represents the same relationship, but we write it down in a different way Now I will give you some The interesting properties of algorithms They are all in fact derived from this relationship From the rules of the regular foundations The first thing is that algorithms … let me use A more pleasant color Logarithm for any basis – let us call Basis – Suppose the base b The base algorithm b of a + the base algorithm b of c – and This only works when we have the same basis This is important to remember This is equal to the base logarithm b of c × a Now, what does this mean and how do we use it? It is better to try to apply them using some Well, I don’t know, examples This means that – I’ll switch to another color Let’s use the purple color – purple – I don’t know I don’t know how to say it correctly Let’s allocate this color to the issues Let’s assume that the base 2 logarithm of – I don’t know – of 8 + The base 2 logarithm of – I don’t know, let’s suppose – 32 The theory says it should be equal, if we assume that it is correct Theoretically, this should be equal to the logarithm of base 2 of what? Well, we said 8 x 32 8 x 32 = 240, + 16 = 256 Let’s see if that is true Only using this number, but this is not a proof of theory But I’m going to give you a little bit of intuition, I think, so that You realize what’s going on So if – this– we have used the feature That simple feature I showed you Let’s see if it works So if base 2 is for 8 2 raised to any force whose product is 8? Well, 2 ^ 3 = 8, right? So this phrase here is equal to 3, right? If base 2 is 8 = 3 2 raised to what force give us the result 32? Let’s see 2 ^ 4 = 16 2 ^ 5 = 32 So this part here is 2 ^ 5, right? And 2 is raised to any power whose output will be 256? Well, if you specialize in computer science You will know it immediately That byte can contain 256 values So it is 2 ^ 8 But if you don’t know this, you can calculate it yourself But this is equal to 8 And I didn’t say that because I know 3 + 5 = 8 But I counted it on my own This is equal to 8 But it does not indicate that 3 + 5 = 8 This may sound mysterious to you, or it may seem obvious And for those who seem somewhat clear Try or think with me, 2 ^ 3 x 2 ^ 5 = 2 ^ 3 + 5, right? It is simply a rule of thumb What do they call this? Power grouping feature – I don’t know I don’t know the names of things This equals 2 ^ 8, 2 ^ 8 And that’s exactly what we got here, right? On this side, we had 2 ^ 3 x 2 ^ 5 And on this side they were combined What makes logarithms interesting and why – it is something Confusing at its start You can see the proofs if you like Verification – Brahini does not provide verification But if you want a better explanation On knowing how this works But I hope this explains why This property is valid, right? Because when you multiply two numbers for them The same basis Two expressions have the same basis You will be able to gather the foundations Likewise, when you have two numbers multiplied by For each other, this is the equivalent of all Of the two numbers sum together It is the same feature If you don’t believe me, see the proof of offers Let me show you – let me show you another feature of the algorithms It’s pretty much the same I consider them the same So this is if the base b is for a – if the base b is for c = If the base b is for – you run out of space I have no space – a ÷ c a ÷ c And again, we can try it with some numbers I used the number 2 a lot because 2 is an easy number Find his strength But let’s use another number Suppose if base 3 of – I do not know – if base 3 of Let’s make it interesting – if base 3 is for 1/9 – if base 3 is 81 This feature says – the same idea Well, I will end up with a large number If the base 3 is for 1/9 ÷ 81 This is equivalent to 1/9 x 1/81 I used two large numbers in this example, but we are We will continue lets see 9 x 8 = 720, right? 9 x – correct 9 x 8 = 720 So this is 1/729 So this is if the base is 1/729 So what – 3 raised to any force whose output will be 1/9? Well, 3 ^ 2 = 9, right? So 3 – we know that if 3 ^ 2 = 9, then we are We know that 3 ^ -2 = 1/9, right? A negative sign means that we flip So this equals -2, right? Then – – 3 to any power it will have 82? 3 ^ 3 = 27 3 ^ 4 We have -2-4 = – Well, we can Calculate this in several ways -2 – 4 = -6 Now we have to make sure that 3 ^ -6 = 1/729 This is my question Is 3 ^ -6 equal to 7 – 1/729? Well, that’s 3 ^ 6 = 729, because the negative exponent He works on his heart Let’s see We can hit that, but this has got to do with the situation Because, well, we can look here lets see 3 ^ 3 – this will be 3 ^ 3 X 3 ^ 3 = 27 x 27 This seems easier You can check it using the calculator if You did not come across me Anyway, this is all the time we have to offer Next time, I will explain to you another Two properties in logarithms If we have more time, we will do some exercises The remaining time see you soon

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