 Hi, I'm Zor. I would like to continue solving problems with logarithms. These problems are also simple. So just try to solve them themselves first and then listen to this lecture. So that's probably very important for you to try to do it yourself first. But anyway, the problems are very simple. So let me just go straight ahead with whatever I've had. The first series of problems is related to equations which is necessary to solve. So log base 3 of 3 plus x equals 2. Now, all these equations are basically illustrations to definition of the logarithm. So with definition of logarithm, well, you know that if base is raised into the value of this logarithm, which is 2, you will have the value under the sign of logarithm. So from here, in using the definition of the logarithm, we can derive that 3 plus x is 3 to the power of 2. So 3 plus x equals 9, x equals 6. That's the solution. And indeed, if you will substitute x here, you will have 3 plus x, which is 9, and logarithm 9. But the base 3 is 2 because 3 squared is 9. So all of these examples are on the same level of difficulty. I'm just thinking that the more equations of this type we will solve, well, the better you will feel about the logarithms and the more fluent you will be in this language. Next, log base 1 half of x over 2 equals minus 3. Again, using the logarithms definition, the base 1 half raised into the power minus 3, which is the value of the whole logarithm, should be equal to whatever the value of under the logarithms is. So 1 half to the minus 3. Minus means it should be inverted. So it's 2 to the power of 3, which is 8, 8 equals, and x is equal 16. And again, if you substitute 16 here, you will have 8. And logarithm of 8 base 1 half is minus 3 because 1 half to the power of minus 3 will give you 8. By the way, I don't think that all problems with logarithms are that simple. These are just illustrative problems. There are much more complex ones, which probably we will address in the next lecture or so. Log of log base 1 10 5 x equals 3. Okay, let me be a little bit faster in this case. 1 tenth to the third degree is equal to 5 x. This is 1 thousandths equals 5 x. x equals 1 over 5 thousandths. That's the solution. So as you know just now, saying less words and more actions because it's actually, I've already said all the words. Everything is supposed to be understood right now from just whatever I'm writing about. Log 2 x to the tenth degree is equal to minus 10. So 2 to the power of minus 10 should be equal to x to the power of 10. Well, it's easier if you will do it this way. 2 to the power of minus 10 is 1 half to the power of 10. And considering that exponential function is really... Well, the graph... I'm sorry, this is not an exponential function. But the graph of this function x to the tenth degree is something like this. It looks like a parabola, but it's steeper. In this particular case, it's very easy to find one solution which is obvious solution. x is equal to 1 half. But as you understand, x equals to minus 1 half is also a solution because this is an even power. Minus 1 half to the tenth degree is exactly the same as 1 half to the tenth degree. So we have two solutions to this equation. 1 base square root of 3 of 3x is equal to minus 2. So square root of 3 to the power of minus 2 is equal to 3x. Now, this is 1 over square root of 3 to the second degree, which is 1 third. So 1 over square root of 3 square is basically squaring of separately numerator and denominator. 1 square is 1, square root of 3 square is 3. So from here, we have x is equal to 1 ninth, right? All right, done that. I hope I'm not boring with all these very, very similar and very simple equations. But again, I'm just trying to say that the more problems of this you solve, the better you will feel about it. This is the last equation. By the way, just looking at this equation, you already have to understand that x is supposed to be greater than 0 because otherwise the square root would not exist and then 1 over square root prohibits x equals to 0. All right, so 3 to the power of minus 1 half is supposed to be equal to this. Now, this is 1 third to the power 1 half. This is square root of 1 third is equal to 1 over square root of x. Now, we can square both sides and that would be 1 third equals to 1 over x, x equals 3, which is indeed positive value. That's why it fits the bill. Well, that's it about the equations. So, next series of problems is calculation. So, you will try to calculate, obviously without the calculators, certain things related to logarithms. I think this is a little bit more interesting. Log base 3 of 3 square root of 3. Now, how to calculate this? Here is how. First of all, you have to express the variable in parentheses as the power of 3. If you will do that, then obviously logarithm is equal to 8. We know that from theoretical lectures we've talked about this many times because 3 to the power of a would be 3 to the power of a. That's the definition of logarithm. So, all we need to do is to transform the variable in parentheses into base 2 sum power. Well, that's very easy to do. 3 is 3 to the first power. Now, multiply by square root of 3, which is 3 to the 1 half. Now, whenever we have two numbers which are exponents with the same base and different exponents on the top, what you have to do is you have to add up the exponents. So, this is 3 to 1 plus 1 half. I hope you remember this. a to the power of b times a to the power of c is equal to a to the power of b plus c. This is one of the fundamental properties of exponential functions. So, this is equal to 3 to the power of 3 over 2. Which means this is equal to log 3 to the power of 3 2. And that's why it's equal to 3 over 2, the exponent of this. Since the base is the same as this, 3 to the power of 3 over 2 will be 3 to the power of 3 over 2. So, that's the way how we calculate it. The most important is to transform whatever is under the logarithms into the base in some power. And we will do exactly the same in all these cases. So, log 1 half base square root of 2 over 2. What is it? It's log 1 half of... Square root of 2 is 2 to the power of 1 half. Then 1 half is 2 to the power of minus 1. This is log 1 half 2 to the power of minus 1 half. Which is equal to log 1 half. Now, since this is minus, we can invert it and we will have 1 half to the power of 1 half. Instead of 2 to the power of minus 1 half, we have 1 half to the power of 1 half. This is therefore equal to 1 half. Next, exactly the same technique. We will use all these times with all these calculations. Log 1 tenths, 10 to the third degree. All right. Log 1 tenths is 1 tenths to minus 3, right? Instead of 10, we can put 1 over 10 with a minus sign in the power in the exponent. That's why it's equal to minus 3. So, what is this? We have to really convert whatever is in the parentheses in the expression of 2 to some power, right? So, first of all, it would be negative power, right? So, it's 8 would be 2 to the power of minus 3, because it's 1 eighths actually, and 1 over 2 would be 2 to the power of minus 1 half. So, minus gives you 1 second, 1 half, and the power of 1 half gives you square root. Which is equal to, now you have to add exponents. So, it would be 2 to the power of minus 3 and 1 half. And that's why it's equal to 3 minus 3 and 1 half. Oh, sorry. 3 and 1 half, which is minus 7. This is square root of 3. 1 to 7, square root of 3. All right. So, we have to convert whatever is here into the square root of 3 in some power. So, let's think about it. 27 is 3 cubed. But if I want to have the base 1 square root of 3, which is this, right? So, instead of 3, I put square root of 3 square. Now, this expression means you have to multiply the powers. So, it's square root of 3 to the power of 6. So, here we have log square root of 3 of square root of 3 to the 6 and square root of 3 to the 1. Which is equal to log square root of 3 of square root of 3 to the power 7. So, that's equal to 7. Right? One more. Base 1 third of 1 over square root of 3. Well, again, the same thing. 1 third, 1 half. That's what 1 over square root of 3 is, right? This is the square root. And that's why the whole thing is equal to 1 half. Sorry, that's 3. Okay. That was the last one of this series. And then I have the last series of problems of the same kind of difficulty or easiness, if you wish. Well, actually they are much easier. So, we're talking about common or decimal logarithm which is the logarithm base 10. So, the question is, what is this? Now, if there is no base indication, it means it's base 10. Well, you know that log of 1 with any base is equal to 0 because any base to the power of 0 would be equal to 1. Next. Log 10. What is the power I have to raise 10 to get 10 obviously 1? Log 100 is equal to 2 because 10 square is equal to 100. Log of 0.1 is equal to minus 1. This is 1 tenths. So, if I raise 10 to the power of minus 1, I get 1 tenths. And finally, log 0.01, which is 100s, this is minus 2 because 10 minus means it's 1 tenths and then square it will be 100s. And the last one. So, log 0.001 obviously is minus 3 for the same reason. 10 minus would be 1 tenths, cube would be 1000s. Okay, that completes the series of simple problems related to logarithms. Please reexamine them again. I think it's always useful, this type of things. And as usually I encourage you to be registered student, to have somebody as a supervisor or actually you yourself can be your supervisor under a different ID. Enroll yourself or somebody should enroll you into courses and that would enable you to take exams, which is very important. Thank you very much. That's it for today.