 So we're going to talk about logarithms, and we're really going to focus on just common logs. But before we can really talk about logs and common logs, we really need to talk about bases of 10 and some special rules that will help you do things possibly quicker if you know them. So let's start with the first two examples, the 10 to the fourth and the 10 to the second. A really cool thing about bases of 10 is that they will always turn into answers or numbers that have a 1 in them and 0s in them. So if you know that, pretty simple. So let's start with 10 to the fourth. I know, based on what I just said, that it's going to have a 1 and a 0. Because the power or the exponent is positive, the 1 is always going to come in the front. And the 4 tells you something. So since the 4, the power is positive, it tells you the number of 0s that will come after the 1. So the number of 0s. So after the 1 that I've already put in place, there will be 4 0s. So 1, 2, 3, 4. So it turns into 10,000. So really simple. Don't need a calculator. We can do that very quickly. Let's try it with the second example. So 10 to the second. Again, I have a base of 10. I know that because the power is positive, that the 1 is going to come first. So there's our 1. And then we know that since the power is 2, I'm going to have 2 0s that follow. And we all know that 10 to the second power is 100. So a really easy way to do that as well. So that's with positive powers. What if you have negative powers? Well, first off, if you have negative powers, the 1 doesn't come at the beginning. It comes at the end. So I'm going to go ahead and put a 1 at the end of the problem. So again, remember, they always are going to have 1s and 0s. Now, the negative 4 does not tell you how many 0s. It tells you how many places you're going to get after the decimal. So it's the number of places, or number of decimal places. So number of places after the decimal. So I should have a total of four places after the decimal. So I'm going to go ahead and put my decimal in place. And I should have four places. So 1, 2, 3, 4. I have to count the 1. In those places that have holes, I'm simply going to put in 0s. So it's .0001. All right, let's try the last one. 10 to the negative 2. Again, any base of 10 is going to involve 1 and 0s. Since it is a negative power, the 1 is going to come at the end. I know that. And remember that the negative 2 again tells me how many decimal places or how many places after the decimal. So it has 2, so 1, 2. The 1 does count in this. So in place of the missing one, I'm just going to put in a 0. So .01. Pretty simple, but these will help you with common logs. So moving to common logs. A common log can be written as log or log base 10. However, you very rarely see it written as log base 10. It's just assumed that you know that the base is 10. And by definition, the common log of a number is the power to which 10 must be raised to get that number. So for example, we're going to start out simple here. If I have log of 10 to the fifth, I'm going to focus on this number. And you're going to ask yourself 10 to what power will give you 10 to the fifth? Well, obviously the power is 5. So your answer is 5. So 10 to the fifth power will give you back this answer. So the log of 10 to the fifth is 5. Pretty simple. So with the second one, the log of 10 to the negative 2. Again, we're trying to figure out 10 to what power will give me this answer. Will 10 to what power? The power is negative 2. So just understand that a logarithm is simply an exponent. Now working with some more difficult ones, where it's not already written out for you. Log of 1,000. Is not written as 10 to a power. It'll be much easier to work with if I can rewrite it as 10 to a power. So I'm going to keep the log part. And then I'm going to take the 1,000 and see if I can rewrite that as 10 to a power. So it would be 10. And let's think about what power it would be. How many zeroes are following the one? Three zeroes. So it's going to be 10 to the third. Going back again to our rules of base of 10. So 10 to what power will give me 10 to the third? Of course my answer is simply 3. It's just the power that's coming with you. Let's try one more of those. So log of 0.001. I'd like to be able to rewrite this as 10 to a power. I know I can rewrite it because it only has zeroes and a 1 in it. So that's possible. So what I'm going to do is go ahead and keep my log and try to rewrite 0.001 as 10 to a power. Well first off I know it's going to be negative because the 1 is after the decimal place. So I know that. And then all I have to do is count the number of decimal places I have. So there's 1, 2, 3. So it's 10 to the negative 3. So again to answer the problem the log of 10 to the negative 3 is going to be the number to which 10 must be raised to to come back with this answer. So 10 to the negative 3 is 10 to what power? It's simply negative 3. So again logarithms, common logarithms are nothing more the power on the base of 10. Now the last one that we have, we have 55. 55 cannot be rewritten as a base of 10. So the only way for us to work this problem is to use a calculator. So on your calculator you have a log button. If you simply just touch log and then 55 and some of you may have to close the parentheses. And you will end up with, I'm going to make this approximately, I'm just going to take it two decimal places would be approximately 1.74. So you're not going to be able to turn everything into a base of 10. Remember it's only those things that have a 1 with 0s after it or a decimal place with 0s before and then a 1 at the end.