Hey guys. You have already watched the class in which I explained about exponential equations. If you have already watched and understood everything, you are prepared for this class, you will learn logarithms, logarithms are a widely used content, right?
In mathematics classes, for example, in financial mathematics, when I work with compound interest in biology, chemistry, geography and other contexts. So if you want to learn logarithm content, I invite you to watch this class. Let's go.
So, welcome to my channel, I'm Gis and before starting our class. About logarithm, I want to ask. 2 little things for you, subscribe to Giz's channel, if you're not subscribed yet, but I think you're already subscribed, right?
So that you receive all the notifications about the classes I publish and, oh, leave me a thumbs up, which is very important for Giz, Ok? So let's go, when I talk about logarithm, then the logarithm, the abbreviation, oh. Log?
Log, so every time you, you see this abbreviation here, it's. Log, okay? And?
I brought it here for you, then. A Log, okay, because we're going to talk about the definition first so you can see the condition of existence. Then we will do the exercise for you to apply.
It is good? So I have. Here, the.
Log of b here, an a here, an x what is this, guys? This is called, let's go. This b.
His name is. Logging this right. Look what logarithmic name this is called base.
It's OK? And that x? What is the result, right?
It's called a logarithm. So look what we're going to do, look. The log so you don't forget.
Okay, the log. So, we know that. This b here.
For us? It is the logarithm, so you don't forget these names, sometimes there will be a statement that says these names, you need to remember, just like when we study the power. So I have the power, I have an exponent, I have the base, right?
When I study radicals, right? Radiation, I have radical radicando. The index, so it's important for us to know the names, right?
Of the lyrics I'm working on. What is everyone's position? What?
It stays here on the little leg. I say that the little leg of g, which is A, which in this case we are going to exchange for numbers. Then it is called base o.
The base is the same, right? Because it has to be the same, which is the result. And our result?
Is it going to be our logarithm, okay? So, guys, I always. I'll have this one.
Situation here using logarithms on a base. And it will result in the logarithm that is our result, so putting it here in letters, then the log of b in base a will be. Equal to X, when?
C only C. For those who don't know the name of this symbol or have forgotten, it is called C only C. Raised x.
Same as B. My God in heaven, what does this thing here mean, oh, this thing here that I put in, I took it from here, where do I do Caracol, Caracol? What is this, oh?
If you catch this guy here. So, and raising this guy here, you get the value of B o a raised to it's OK? So apply Caracol to find out this base?
Raised to the x? It will give the value of b. Well, A raised to X equals B very well, but then you have to know to apply these concepts here you have to have some conditions, you have to respect the conditions of existence.
What are these conditions? Oh, B people, what is this number? Stay here?
What has to happen to him? It will always have to be a number greater than zero. What does that mean?
That it can never be zero nor can it be negative. Okay, so always the number. Positive, okay, maybe.
Any positive number, me. A? Rules for oa, pay attention, oA.
It has to be greater than zero, right? So, it cannot be negative. And look here, guys, and at the same time it has to be a number other than one, because you'll think, Ah, if the greater than zero can be a non-OA, then, in addition to it being greater than zero, any number greater than zero, it cannot be 1.
It has to be different from 1, right? So mark these conditions of existence here, because we need to remember them well to solve the exercise correctly. Okay, let's do it now, you already.
He knows? How does it work now? We are going to apply this here using the numbers to find the actual value.
Come on, guys. So, lets do it. Now, the.
Application of those initial concepts that we saw, which are the conditions, right? And the definition? Initial, look now.
Which I brought here log(32) in the base. 2 What is the value, then, of the logarithm? So you see you don't have one?
Like here. There was the same thing there, at X. Why?
Because I'll go. Write now with. You, oh, so it is.
Equal? to X. Right?
And now, how do I ride it? Same? Does the resolution strategy here remember Caracol?
The production team said that my Snail looks like a slug, it was all a bunch of clumsy thread, so now I'm going to try to make a cuter Snail, oh, so what am I going to do? 2 to the power of X. It has to be 32.
Oh, I'll do it. Make a slug face here. Now, guys, look at my slug, my slug, not my Snail.
Then, production. Did I say it wrong? Look here at my Snail.
How cute is it now, guys, what's up, right? Now it looks like a snail, right? Yes or no?
So here. Applying Snail. Here, you won't forget.
How do you solve the logarithm? Okay, so here it will stay. 2 to the power of X, this to the power of X has to give o.
32 what? What's that? So, guys, this isn't an exponential equation like that class you attended, where I explained about exponential equation.
If you haven't attended this class, I'll leave the recommendation here, okay, because it's very important to remember this concept. So here I need it. Leave the two bases here, equal to this 32 so I can leave this 32 in a base.
2, yeah. I need to decompose 32 into prime factors, okay? So let's go, 32.
2 is 16, 2 is 8 and 2, 4 is 2, 2 is 2 of 1. But if you already know, know mentally how much 32 is in the base, 2 you don't need to do it here. Like me.
I did it, okay? So I have 12345, so that means that 32 is the same thing as 2 to the power of 5, okay ? The objective was accomplished, wasn't it?
I managed to leave the 2 bases. Right, the powers are equal here. So here it is 2, here it is.
2, then I'll say goodbye to 2? Bases. Okay, and I'm going to work with the equality of exponents, so that's what's left.
Here, X is worth 5. But what does it mean for x to be 5? It means that.
The log (32) in base 2. It's equal to 5. That's the.
Value then, in our logarithm is 5, right? When I have 32 in base 2, OK, let's do the next one here, then too, look. Same as X.
And do the Caracol, guys, you won't forget the other Caracol, ok . Then he makes his face cute. I'll just do it here.
I promise I won't do anything more as a designer after that, guys. Me. I'm wonderful, right?
Look at my drawing of Caracol, how beautiful. So now, what does the strategy look like here? 3 to the power of X is equal to 243.
What do I need to do then? I need to decompose 243, invoicing. Here in cousins.
Get it. 3, which gives 81, gives 3. Which gives 27, gives 3, gives 9.
For 3 gives 3. For 3 gives 1, so it will be 12345 too. Guys, oh production.
They both gave 5 so, here it is. 3 high. The 5 and then I think the production purposely made the 2 gives 5, Huh, guys?
Here then I give. Bye to base. Since they are the same and I can only do this if they are the same, okay?
And I do equality. Among the exponents. So what was x?
It's equal to 5, so that means. What? Log(243) in base.
3 is equal to 5. Okay, guys, let's do some examples. Now, a little bit.
More difficult? Come on, guys, before doing the next examples, right? That I said a little more difficult.
I want you to notice something, because many students are in doubt, Oh, I said that time at the beginning of the condition of existence, right? What couldn't happen? Even when I calculate log, the value of b there, which was the logarithm and I send it to you, What could he not be?
Since b had to be greater than zero, so it couldn't be negative or zero, okay? And the A that would happen to the A had to be the number greater than zero. And different from one?
So many students went, but these days, why does this have to happen? So I brought some cases here now that you already know how to calculate a logarithm, I brought them here for you to see. O.
Why do those conditions of existence have to exist? Oh, then. Let's go.
Apply Snail. Here, here I'm not going to draw Caracol anymore, huh? I already drew mine very nicely.
Snail. Oh, so here, minus 3 to the power of X, has to be 27, that's not what I do. So, look, minus 3.
To the power of a 3. And now how do I end a situation like this? It doesn't work, guys, why doesn't it work, because look here, the base here is minus 3.
Here, the base. It's 3, there's no way I can work now, continue this type of exercise, because the bases are different. So for that reason, that was exactly who this minus 3 was.
The base wasn't, so that was the reason that the base had to be a number greater than zero. It couldn't be negative, so one of the conditions is that A had. What to be greater than zero?
So here we are showing you why it cannot be a number less than zero, which is the negative number. OK? Now let's see.
Here. O. Zero, which is our base, so I'm going to do Caracol.
Equal to. X, zero to the power of X has to be 5. Come on, zero to the power of There will never be a zero raised to any number here that gives 5, it also doesn't work here, even if it's not possible to do this calculation here and it's not impossible to do it here either.
So what happens for this reason that a has to be bigger? What a zero. Huh, but.
It applies here, here it is. Here, O greater than. Zero, I brought it.
To dispute the number with you. Negative here, for. Prove to you that it doesn't work.
I brought exactly zero, o greater q. If had. That little line here, guys, with that little line, would that indicate that A had to be a bigger number?
Or. Equal to zero, then it could be zero, but notice that I don't have it. Then the.
Second condition of a. And what was the other condition? It's not the second condition, right?
I wrote these 2 here as one, the one greater than zero means what? That it cannot be either negative or zero and the other condition is that A could not be 1, it had to be different from one. Let's apply it here.
Same as X, oh the Snail. So it will be 1 elevated. to X, is equal to.
3. So, one to the power, how much does 3 give? Do it for me?
This calculation is not possible, there is no way to find the number that I raise here, the one that gives 3 then it is another calculation that is impossible to do. It is good then. There, that's why.
Explanation of the condition? Do A ter? That being different from 1 is right, oh, so the conditions here of A, so, these accounts here, oh, are not possible to be.
Be carried out? OK, now let's see what was b? I put x here and apply Caracol, so 2.
Raised to X, it has to be. Equal to minus 4. This is going to happen.
The same context as here, oh, because at the time. That I? Carry out the decomposition.
Oh, it will be 2 raised. to _ _ So, that's why B had to be what, guys? the B?
the B had to be bigger. What a zero. And then, it falls into the same context here as this one, A greater than zero.
It can't be zero, did you see I put zero here? It didn't work, the same thing will happen. With B.
Oh, you want to see the log. It has to be bigger. It can't be zero of zero in base 5.
It has to be. Equal to X, oh. I apply curl, oops I made a mistake.
Snail, guys, Oh, my God, I'm so excited about my Snail, he was so. Beautiful, oh. Log, then.
Applying Caracol 5 to the power of X, it must be equal to zero. Now tell me 5 to the power of zero. It won't work either, right?
So I can't do this calculation here either, okay? So, that's why the B. It has to be greater than zero.
It also can never be a number equal to zero. So, I brought it. For you here, these explanations so you can see why there have to be those conditions of existence, right, guys, now we're going to make those examples a little more difficult.
Let's go. Guys, look at the next examples, a little more difficult, look. So, for you to perform well in these exercises, it is important that you remember the concepts there.
Properties? Of the powers, okay? I will leave the indication for you here and it will also be in the description, because then you access these classes and remember these concepts that are important to apply here?
OK? So, look here, let's do ours. Dear Snail.
7 to the power of X. It has to. Fraction 49 is our loraritimand, okay, guys?
Here it is, Gis' writing, it didn't look very good there, right? So here I have 7. To the power of X is equal to 1/49.
And now, how am I going to transform a fraction into the base? 7, my God in heaven, is that right? We haven't verified the conditions of existence , have we?
I just put it in and did it, so it's important to check , the A greater than. Zero different from one, OK? o B.
B greater than zero, because it can be a fraction, it can be a decimal number, so that's right, so everything is OK. So does that mean there's a way to get out of this? Yes, what now?
That's why I said you have to remember those concepts. Properties of. Powers, okay here, guys, first thing, if I calculate this 49, 49 is the same thing as 1/ 7 raised to 2 because 7 × 7 = 49 practice this here, right?
I don't need to do factoring. And now, how am I going to make this 7 go up? Ah, then there was that exponent process where when I invert the base, I change the sign of the exponents.
I still said in class, oh, it's going to be inverted, oh, I had to do the whistling for it to be worth it. It's OK? So here it will stay.
7 to the power of X equals. So this 7 raised to 2 will rise, it will change places with the 1, it is the inversion. From base, e.
g. The 1 will go down, but this 1 here remains in the denominator. You won't even need it because any number divided by one is itself.
Right? So, what happens to the exponent? The exponent flips the sign.
He was positive at the time. I inverted the base, it became negative, so do you want more details about this explanation of inverting and changing the exponent sign? Go back to class.
Properties of. Powers, okay? And now, oh base.
I also say goodbye to the bases. What's left? Equality between the exponents, then, in this exercise.
It turns out that x is equal. Minus 2. So that means.
That the log of 1/49th in base 7 is worth minus 2, OK. And this one here, folks, let's equate it to. X?
Let's check first to see if it's possible to continue, shall we? Why isn't it? Base.
It has to be greater than zero and different from one, it's OK. Logging almost like I said rooting, right? I don't usually talk about roots.
The logarithm is a number, here it is a number that is zero. So, okay, let's continue. Snail, then it will stay.
8 to the power of X is equal to root. Cubic size of 16. And now guys, how do I do?
Do it first, can I transform this 16 into writing with a base 8 power, right? With base 8. It’s not possible, right?
Because 16 is half 16, which is 8. But when I do impotence, it's not half, it's the number times itself. So here.
Hint, it's you. Do the decomposition of 8 and the decomposition of 16. Oh, let's do it here then, decompose 16, do a decomposition of 16, look.
It gives 2 which gives 8 to 2, it gives 4 to 2, it gives 2 to 2. It gives 1. Guys.
So does that mean 16? Here. Oh, I'll write it like this, oh.
Cube root, o. 16, is 1234, is 2 to the power of 4. So, it's in 16's place, wow, almost.
Which looks like. 24, right, guys? It's 2.
High, the fourth. It's 8. Guys.
It's like decomposition, you can look here now, oh 8, it's here, oh. From there, what is the decomposition of 8, oh 123. So 8 is 2 to the power of 3.
But it is not raised to the x people, so I'm going to put a parentheses here, oh. Raise x. Right?
Good, oh. Perfect. So, base 2.
Base 2, but it's not that perfect yet, why? Because this 2 is inside the root and here there is no root, so I will have to remove that 2 from the root. How do you do this?
Good. Here, to remove this parentheses, I multiply the exponents. So here goes.
Get 2 to the power of 3 x. You could have done it straight away and you didn't even need to write relatives, okay? But just to explain, in detail.
Same now. To remove this from the root, guys, when I remove a radicand from the root, right? Write it out.
The exponent will be in the form of a fraction, so here it will be a fraction, then it will look like this, oh this guy 3 here, oh, it's a guy who was in the sun, let's think like this, oh. Look at my drawing here, look. 3 was here and the sun is here, oh.
3 was in the sun, so whoever is in the sun wants to go in the shade, so if 3 was in the sun, he will go in the shade, right? And 4? 4 wasn't cute in the shadows, right?
Because, oh, it was 2 to the power of 4. Oh, he was cute in the shadow, so who's in the shadow? It goes to the Sun.
Oh, it's on top, so it's going to be in the Sun, just a way for you to decorate, okay, guys? Or could you think so, oh? Who's for?
Inside, it's on top, who's on the bottom is it? Is the outside good? Oops, I said it backwards.
Whoever is at the bottom is at the bottom. Outside, it's fine. Then?
Either way, you can memorize this concept of taking the number from the root and its exponent will be a fraction, okay? Well, look here now how cute it is, look, the base looks the same, goodbye to the base. There was equality between them.
Exponents, so 3 Times x is equal to 4/3. And now, what about me? I'll do it from here.
Oh, this 3 isn't multiplying here, so I'm going to divide it, but the other way is faster, we can do it too, look at this 3, let's look at. Here's the side. This 3 here is not being divided by 4, so since I have an equation, I.
I'm going to take this 3 here and throw it over there. It will go there multiplying. So it will be 3 × 3.
Will it be 3 × 3? 9x? 3, × 3.
9x equals 4? So now, to finish, it will stay. X, equal to 4.
This 9 I was multiplying, comes dividing. So, in this question of logarithm, the log(∛(16) in base 8 will be? Equal to 4/9, right, guys?
So, an exercise a little more difficult, but because a little more difficult, because it demands it. But the application of the properties of powers, right? And then I'm going to end the class with these examples here, okay?
So this is considered class 1 of logarithm, okay? We then made the definitions, the conditions of existence. You didn't forget no, right?
Is the wonderful Snail that Gis made ok? And we did these examples here, in the next class I'm going to continue the sequence of logarithms, so be sure to watch the next class and make it. That request.
For you, if Sign up. On the Gis channel, because then you can receive notifications. About classes?
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And until next time class, bye. . .
