 This video will talk about logarithms and logarithmic functions. So an exponential and logarithmic functions are inverses of each other. Talked about inverse functions, so the x and y coordinates of the exponential function are interchanged to make an order to pair for the inverse logarithm function. I mean the domain of the exponential function becomes the range of the logarithmic function. So the domain of an exponential is all reals, so now the range of the logarithm becomes all reals, and the range of an exponential function is y is greater than zero, and now the logarithmic domain will be greater than zero. That just means that it's not going to cross the y-axis, like that exponentials don't cross an x-axis. Let's see if I can draw you a little picture here. So we have a spread one will be the exponential function, where it doesn't cross the x-axis, and this blue one will be the logarithmic function, and it doesn't cross the y-axis. It can cross the x-axis, but it won't cross the y-axis. So for positive numbers x and b, not equal to one, I have to remember that, then this y is equal to log base b of x, if and only if, as that one-to-one function, x is equal to b to the y. So the function y equal log base b of x is a logarithmic function with a base of b, and the expression is simply called a logarithm, it represents the exponent on b, that's important, represents the exponent on b that gives us x. So we have a couple of logarithms that we can use in our calculator, and otherwise, we're going to need some other way to work these things if there is another base. So common logs are base 10, that's what our number system is based on. Remember when you were younger in elementary school, you learned to add your ones column, and your tens column, and your hundreds column, and your thousands column, because those are all 10 to the first, 10 to the second, 10 to the third. So those are all base 10. So we call it a common log, and when you see log base 10, you don't usually see that. You just see log 10, and you know that it's assumed that it's base 10. And we have natural logs, and these are base e, and e is a natural number that happens in science and in nature. It has a value, and I don't know how to tell you to emphasize it enough, but to tell you that e is a number, it's 2.71, some blah, blah, blah, blah. Okay, it's an irrational number, it goes on and on and on forever, but it is a number, it's not a variable. So we don't use your right log base e of x, because it's this naturally occurring number, we can use ln x. Let's see if we can do this conversion of logarithmic to exponential function. And remember, up at the top, this really looks like a y, and this looks like an x, and this is our b, and so when we want to write it as x is equal to b to the y, we would say x, which is 9, is equal to b, which in our case is 3, to the y, which happens to be 2. Let's try the next one. ln e is equal to 1, and this is one of those natural logs, so it's an assumed base e. So don't think that this is your base, it is, but it's also what that's the artist called the argument. So this is going to be y, and this is x, and then b is going to be e, because it's a natural log. So we have e to the y, which is 1, is equal to x, which is e, and you can see that that's definitely true. If we see this log 1,000 equal 3, again, this is my y, and this is my x, and then my base is not here, so we know that the base is 10. So we would say b to the y, which is 3, is equal to x, which is 1,000, that should be 1,000. So let's move on and look at, we want to go the other direction, write each equation as a logarithmic function. Well, if you go back, it said log base b of x was equal to y, and then it was also b to the y is equal to x. So we just need to figure out what's y and what's x. Base is hopefully obvious. Here, b should be e, and we have our y, which is our exponent, so y equal 3, and then x is going to be that 20.086. So to convert it, we write log, because we are an exponential, so we're going into a log, and we don't really want log. I forgot what my base was. My base is e. I don't know why I put 3 there. My base is e, y is the exponent of 3, x is what is equal to 20.086, and since it's base e, I'm going to say the natural log of e, but I don't have to write a subscript. Okay, so I've got this part, this natural log of base b is already taken care of. Now I need to put my x in, and x was equal to that 20.086, and then it's equal to my y, which was 3. Again, b is equal to 7, y is equal to my exponent, this is the form I'm in, so this is 0 is y, x is equal to 1, and I need to write it as log, and this time it's not 10 or e, so I'm going to write log, and then it's subscripted 7 so that we can realize that that's the base. Log base b of x, which is 1 in our case, is equal to 0. Try again. This time we have base is equal to 20.07, and it looks like our y, the exponent, is equal to 2 thirds, and it looks like x is equal to 9, so log base 27 of x, which happens to be 9, is 2 thirds. And if you remember, we were talking at the beginning of this video about, this is read, the log base b of 9, and we're trying to find the exponent on 27 that will get us 9, and 2 thirds, x 27 to the 2 thirds would be 9, just like 7, the example above, 7 to the 0 is equal to 1, we know that's true. So logs we're finding out, and exponents have a relationship. So what we've really found here is that they not only are related, they are the same thing. Logs are equal to exponents.