 This video is going to talk about logs in an introduction. And the main thing you need to remember is that, so logs are exponents and they are a number. If you can keep that in mind, that'll make logs a little bit better. Now when you see log that looks like this here, it's red log base b of x. So the base is going to be some little subscript number down here. And then you're going to have this x, which is actually called an argument. So y equal log base b means that y over here on this other side, remember it logs an exponent. So y is the exponent on b, the base, that will give you this x. So we want to simplify these logs. And remember it translates to find the exponent on the base that makes this statement true. So if I have log base 2 of 8, I'm really saying 2 to the what is going to be equal to 8. 2 times 2 times 2 is 8. So 2 to the third. Log base 2 of 8 is equal to 3. That's the exponent on 2. So again over here, log base 3 of 81 is what exponent on 3 will give me 81. And if you don't know, you could always use your calculator. You come in here and say, okay, well 3 karat x. Now if we look at our table and we're looking for 81 in this y column, and it looks like it's 4. Right here is 4. So we would say that log base 3 of 81 is 4. Notice I found that log by actually thinking about the exponential. Let's look at this one. Log base 5 of 625. So again, not 6, 5 to the something is going to be 625. Well, I know that 5 to the third is 125. So I would guess that 5 to the fourth is going to be 625. And I could always take 125 times 5. This would be the long way to go, but carry the 2. There's 10 plus 2 makes 12. 5 plus 1 more makes 6. So sure enough, 5 to the 625 is, or 5 to the 4 is 625. That's the ways you can figure these out. Sometimes you just know them. Sometimes you could use your calculator. Sometimes you could use what you know to help you get to the next part. 4 squared is 16. Doing it the long way again. If I multiply that by 4, I'm pretty sure that's going to be 64 because 6 times 4 is 24. Carry the 2. 4 plus 2 more would be 64. So I had two factors. I got one more. So 4 to the third must be 64. So you can see that now we're going to convert a log into an exponential. You can go back and forth both ways. What you're really doing is switching the inputs and outputs. So here in the log, this would be our input. And the y would be our output. And over here, you can see that the y, which was our output, is now our input. And the x, which was our input, is now the output. Let's think about what the input and output are. The input here is x, and the output here is 7. So when I go to do it in the other form, in the exponential form, my input is going to be 7, and my output is going to be x. Now the thing we need to remember is that both of them have the same base. We're using the same base for just using a log or an exponential. So I have a log, and this is my base. So I have a base of 2. And on an exponential, we input in the exponent. So we're going to input the 7, and then we get an output of x. Let's try that again. Over here, for the log, my input is 36, and my output is 2. And when I do the exponential, my input now is going to be my output. So that will be my 2. And my output will be what was my input. So that will be 36. And then the only other thing I need to know is that b is equal to, in this case, x. So we have our base of x, our input, and the purple over here. And the exponent is the 2, and then it's equal to our output, which is 36. There's another way that you can do it. If you just want to, if you're a visual person, you follow the next way that you'll see a visual way to do it. You can also express these, like we did on the first slide, by converting it. Log base b of x is equal to b to the y equal x. I always say it's like hopping across the pond. You take the base, because we know exponential starts with bases. You hop across the pond to get the exponent. Remember that's the exponent over here. And then you hop across the equal sign to get to the other side. So you start with your base. We hop to cross to find our exponent. And then we hop back across to get to what it's equal to. So 2, hop across the pond, gives us 2 to the 7th. And that's going to be equal to this x. So 2 to the 7th, we could just go right to our calculator with that. And just go into our home screen. 2, care at 7, that's something we can compute. And 2 to the 7th is equal to 128. So x must be 128. In other words, log base 2 of 128 would be equal to 7. This is our answer right here. But the true statement would be log base 2 of 128 would be equal to 7. Let's convert this one. We start with our base, that's x. Hop across the pond to get the 2. And then that's equal to hopping back across the pond. And we're back on the other side. So we have to be on the other side, equal to 36. And now this is a variable, now is our base. And we can take the square root of both sides. So the square root of x squared is going to be x. And the square root of 36 is going to be 6. Now, some of you might be saying, isn't that supposed to be plus or minus 6? Well, remember about bases. The base has to be greater than zero. And so we can only accept the x equals 6. Let's try again. Start with your base. Hop across the pond. Negative 4 is equal to hopping back across the pond, the x. The 5 to the negative 4 is 1 over 5 to the 4th. And I think earlier we found out that 5 to the 4th was 625. So using that fact now, we have 1 over 625 would be equal to x. And that is a perfectly fine answer. x raised to the 3rd is equal to the 8. If you can just keep reminding yourselves that the other side of the equal sign is the exponent, then this conversion can be a little bit easier. Again, my base now is my variable, so I can take the cube root of both sides. This gives me x on this side, and the cube root of 8 is 2. Or if you needed the calculator, you would say go to math. Cube root is 4 of 8 is equal to 2. We can also calculate some logs. If they are common logs, that's base 10. That's what our number system is based on. We've got our ones, our tens, our hundreds, our thousands. We're on base 10. So that's why we call it the common log, because that's our number system. So we have a log key, because we use it so often. And then base e is 2.718, blah, blah, blah, blah, blah. It's a natural number. It's called the natural log, and it's ln. That's not an i. It's an l. And in Latin, this would be logarithmic naturalis or something like that. So that's why it's ln. The l comes first, in case you care. And so we can just calculate these right from our calculator. Log 60. Notice there's no subscript down here, because we know it's base 10. So just log in our calculator, and then 60. And log 60 is equal to, I lost it, 1.78, approximately. So approximately 1.78. And we can do the same thing with the ln. Now the ln is below the log key, and it's ln 3.5. We get 1.25 approximately. So now we know how to figure out logs. If they're logs or natural logs, we can go to our calculator. If they're equations, then we can convert them into an exponential and go from there. Or if they're just asking us to evaluate a log, we just have to think, what is the exponent on our base that'll give us this number, which again is called the argument?