 This video will talk about solving exponential equations. When we were doing log equations, remember we had to isolate and get one log, a single log on one side. We want to do the same thing, we want to isolate the exponential term on one side and everything else on the other. And we've done some of these with light bases, but now we're going to come across problems where we can't really figure out what the same bases are. So if an exponential is base B, which is not 10 or E, then logs of the base can be used along with the change of base formula. So here we have a problem where we want to get to this exponential all by itself. So we need to add the 3 to the other side. So we're going to have 9 to the 3x is equal to 78,465 when I add 3 to both sides. And now I need to take the log with the same base. So log base 9 of 9 to the 3x is equal to, and I took the log base 9 on that side, I'm going to take the log base 9 on this side of 78,465. And this log and the base are inverses, so they undo each other and I have 3x is equal to log base 9 of 78,465. And then we have this case here now where we can convert this and change the base on it. That would give us the most exact answer that we could put in our calculator around if we wanted to. So we can say that this is 3x is equal to, remember it's the log of the argument, so log 78,465 over the log of the base, which is log 9. And then for our final answer then we just have x is equal to, and this whole thing divided by 3, and this is what we call an exact answer. We haven't rounded anything, it is an exact answer. So what are we going to do here when we have two bases but they're not the same? We can take any kind of log we want, it doesn't make sense for us really to go to log 7 or log 4 because we don't know anything about those. And we would have to convert later anyway, so why don't we just take a log. And we can take a log or a natural log. Some people like to do it, set it up as ln 7x is equal to ln 4 to the 2x minus 1, and some people would rather do it as the log of 7x is equal to the log of 4 to the 2x minus 1. It doesn't matter which method you use. Some people like to just use ln and then when they get to e they don't forget. So let's just do this one with an ln, we've done so many with logs before. Let's do this one with ln. So here is the equation I'm going to use, and we have the property that says log of a raised to the c is equal to c times the log of a. So we're going to use that property here. So x times the natural log of 7 is equal to, in parentheses, since more than one thing in that exponent, 2x minus 1 times ln 4. I got an x over here, but I also have an x in here. So that means that I need to distribute my ln 4 to both of those things, so I can get that x term out of the parentheses. So over here, I have x times ln 7, and that's going to be equal to, and then when I distribute, I will have 2x times ln 4, and then I will have minus 1, or just minus, if you prefer, ln 4. Now I've distributed, so I need to collect the x's. I will move this one to the other side. I'm going to subtract it. I will end up with x times ln 7, and then minus my 2x ln 4, and that will be equal to, on the other side, that negative ln 4. I have an x term here, and I have an x term here, and they have a greatest common factor of x, so I can factor that out. I have an x, and what do I have left over? Well, in here, I'm going to have, here's my common factor. So what's left is ln 7, and then here is my common factor of x, so what's left is minus 2 ln 4, and then on the other side, we still have minus ln 4. And then our final answer will give us that x, and that's going to be equal to what we had over on the other side, negative ln 4 divided by, and what do we have in blue down there? We have ln 7 minus 2 ln 4, and that would be our final exact answer. Okay, now take some time and try the next problem before you watch how the solution happens. Again, we have different bases, and this time we do have a base of E, so I definitely want to take, in this case, the natural log of both sides. And then we want to apply the property that says log A of C is equal to C, times the log of A. Again, we have parentheses here with an x inside it, and here's an x. So I need to distribute into my parentheses. I want to keep my x as positive if at all possible, so I'm going to take this term and go that direction, this term and go the other direction, which the x ln E stays on this side. And then I end up adding x ln 3, and that will be equal to, when I add it's 2 positive 2 ln E. So again, what did we do up here? We collected the x's on one side, and over here now we are going to take the greatest common factor. So we have a greatest common factor of x ln E plus ln 3. Cuz again, here's my greatest common factor, so that's on the outside. And then that will be equal to the 2 ln E. And so for a final answer, make sure that we can always see it. We keep our x, but we're going to divide everything off of it. So x is going to be equal to our 2 ln E divided by ln E plus ln 3. And I probably should have done it a lot earlier, cuz we have one more step. We have to remember here, the ln E, the exponent on E that will get me E is 1. So everywhere that I see that I had ln E, it's really just 1. This is a 1, and this is a 1. So my final answer should really be x is equal to 2 over 1 plus ln 3. That's the final answer.