Introduction to logarithm properties (part 2) | Logarithms | Algebra II | Khan Academy

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Welcome, I will now show you the last two algorithms The first I always find it most obvious But do not feel bad if you are not Perhaps it will take some time to explain it I encourage you to try all of the characteristics of these algorithms Because that is the only way for you to learn it The goal of mathematics is not just to succeed in your next exam Or obtain a degree of excellence in the exam Rather, the goal of mathematics is to understand it You can apply it in real life You will not have to re-learn everything every time The next logarithm property is If I have an A x logarithm of base B for C, that is, if I have A x all of this So this is equal to the logarithm of the base B of C ^ A Amazing thing Let’s see how to do that Suppose I have a 3 x base-2 logarithm of 8 This feature shows this The logarithm will equal the base 2 by 8 ^ 3 This is the same Well, that’s equivalent – we can find it Let’s see what this results from 3 x if the base – what is the result if the base 2 of 8? The reason for my hesitation just a moment ago Because every time I want to find something I would implicitly use the rules of algorithms and foundations to find it I will try to avoid that Anyway, let’s get back to the issue what is it? 2 raised to what force give us 8? 2 ^ 3 = 8, right? So 3 We have these 3 here, so 3 x 3 So this should be equal to 9 And if it equals 9 Thus we know that this feature works at least for this example You don’t know if it will work for the rest of the examples So you may want to take a look at the proof that we did in other shows But this is an advanced topic But the important thing to understand first is how to use it Let’s see, how much is 2 ^ 9? Well, the result will be a large number Actually, I know what –256– Because we are in the last show, we found that 2 ^ 8 Equals 256 So 2 ^ 9 should be 512 2 ^ 9 should be 512 If 8 ^ 3 also equals 512, then we’re right, right? Because if the base 2 of 512 equals 9 What is the output of 8 ^ 3? He’s 64, right? 8 ^ 2 = 64, so 8 ^ 3 – let’s see that 4 x 8 = 32 6 x 8 – looks like 512– True There are other methods that you can follow Because you can say that 8 ^ 3 Equivalent to 2 ^ 9 How do we know that? Well, 8 ^ 3 = (2 ^ 3) ^ 3, right? I rewrote the 8 We know by the laws of foundations that (2 ^ 3) ^ 3 Equivalent to 2 ^ 9 And in fact it’s a grounding property, where you can hit – When you raise the number of one power and then raise the amount as a whole, to another power In this case, you can hit the foundations– This is the property of the foundations that lead us to this logarithms feature But I will not focus much on this show As there is a full review about this evidence in particular The following logarithms feature that I will explain – Then I will review everything and maybe we will solve some examples Maybe this will be the most useful logarithmic feature if you are a calculator addict I will explain why Suppose I have a base B of A = If base C for A ÷ if base C for B Now why is this feature useful if you are a calculator user? Well, let’s say you went to class, and you had a test The teacher said that you can use a calculator Using your calculator, find the base 17 for 357 In this case, you will be pushed and looking for the Base 17 button if your calculator is on You will not find it Because there is no button for the base 17 algorithms on the calculator Maybe you have a button if Or the ln button As you know, if the button is on the calculator It is the basis 10 The ln button on the calculator The basis is e For those who are unfamiliar with e, do not worry about this It’s 2.71 Any number It’s an amazing number, but we’ll talk more about it in another show But there are only two bases available on the calculator If you want to find algorithms another basis Use this feature If you give this on the exam So you can confidently say, oh, well that’s equivalent You have to change to yellow in order to deal with confidence. Basically – we can use e or 10 We can say that this is equivalent to the base 10 for 357 ÷ If the base 10 is for 17 You can print 357 on a calculator And you press the Lo button And you will get some value Then, you know, you can erase it Or if you know how to use parentheses in a calculator, you can do that But later you will type 17 on the calculator You hit the Lu button, and you get a value Then you divide them, and you get the answer So this is a very useful feature for those who use the calculator a lot Once again, I do not want to go deeper For me, this is most useful But not quite It does not come from the characteristics of the foundations But it is difficult for me to describe intuition simply Perhaps you would like to see proof of this If you do not believe the reason for this to happen But anyway, with all this Perhaps this is a feature that you will use often in your daily life I still use it in my work As you know, logarithms are useful Let us solve some examples Let me rewrite a bunch of things in simplified pictures If I wanted to write if the base 2 of the square root Let me think of something. For 32 ÷ cube root – no, I’ll just take the square root ÷ square root of 8 How can I rewrite this in a non-messy way? Well, let’s think about this This is equivalent to, or is equal to I don’t know if I’m going to move vertically or horizontally I’m going to go upright– This is equivalent to base 2 of 32 ÷ The square root of 8 ^ 1/2, right? We know by properties of logarithms, the third property that we have learned That’s equal to 1/2 X The logarithm of 32 التر the square root of 8, right? You took the exponent And I made it a treat for everything We have learned this since the beginning of this show Now we have a division outside here, right? The logarithm of 32 ار the logarithm of the square root of 8 Well, we can use – Let’s keep the 1/2 out This is equal to (logarithm) – Oh for registration I forgot the basis The base 2 logarithm of 32–, right? Because this is beyond division – The base 2 logarithm of the square root of 8 True? lets see Well, we have a square root again here We could say this is 1/2 x if base 2 is 32 – 8 ^ 1/2 Equivalent to 1/2 if base 2 is for 8 We learned this from the property at the beginning of this presentation Then if we want, we can distribute this original 1/2 This is equal to 1/2 if base 2 is 32 – 1/4 Because we have to distribute that 1/2 – – 1/4 if the base 2 is for 8 This equals 5/2 – 3 3 x 1/4 – 3/4 Or 10/4 – 3/4 = 7/4 Perhaps I made some logarithmic mistakes, but you undoubtedly understood the idea see you soon!

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