 This video is part 4 of solving equations, and this time we're going to take exponentials on logarithms and just mix them up. So we start off here with a logarithmic equation, and if we remember with logarithms, we have a couple of options. We could change the base on it, since it's already in log form, changing the base is kind of nice. Or we could convert it to an exponential. Or if we could graph it, but remember we can only graph if it's log 10 or a natural log. It's a common log or a natural log. We don't have a common log here, we have base 4. So we have to choose one of the other options, and I'm going to choose change of base, just to give us one more practice at it. So remember with the change of base, it says log base B of X is going to be equal to the log base C of X divided by log base C of the old base. So this is our X, and this is our B, and we're going to change the base on just this side. So we have log of our new, we're going to go to the common log, so we'll just write log of the inside, which was our argument here, that other number, 64, divided by the log of our old base, which we said was 4, and that's equal to X. So we're just changing the base on this side equal to everything else on the other side. And now we're actually done. Now remember this is not the log of 64 divided by 4, that's a common mistake. Log 64 is a number divided by log 4, which is another number, and this is the exact answer. If I want the approximate answer, I could say log 64 divided by log 4, and we would have that it's approximately equal to 3. In this case it actually is equal to 3, but this would be a good enough answer. They're both exact, but you don't necessarily know that by looking at that, so I would take either one. Now we have an exponential equation. And remember with exponential equations we have the choice that we can, we have to get B to the X by itself, and we have to do that first. But then we have choices. We could convert it and change the base, or we could take the log or natural log of both sides, or we could graph. Well I'm going to choose this time to take the log or natural log of both sides, but we have to get X by itself first, B to the X by itself first. So this is what I'm going for, and to do that this 10 is the farthest away from that 5 to the X. 2 is right next to it. This has got plus 10, so it makes it farther away, so it's an outer layer. So we have 2 times 5 to the X is equal to 16 minus 10 will be 6. Then we divide both sides by 2 because we have to get again this 5 to the X by itself. Now we have 5 to the X all by itself, and 6 divided by 2 is 3. I'm going to take the log or natural log of both sides. I usually take the log, so this time I'm going to take a natural log just to be different. So the natural log of 5 to the X, remember it doesn't matter which kind we take, is equal to the natural log of 3. And the property, we're using the power property here. And the power property says that we have to take the exponent and it can become the coefficient. So X times the natural log of 5 now, because that X has been moved, is equal to the natural log of 3. And if we divide by the natural log of 5 on both sides, then on the left hand side here, we're going to have X. And on the other side, we'll have that natural log of 3 divided by the natural log of 5. That's our exact answer and we can call it done. Now what if we have a problem like this? Remember this is an exponent that has a involved looking exponent. But we just say that that's going to work as one unit. And we can either, again, it's already isolated, so we can either use the power property, that's taking the log of both sides, or we can graph it, or we can change the base. Change the base may not be the easiest thing to do here since it's already an exponential. We'd have to go to a log. So that may not be the preferred method. It might be better to either use the power property or to graph. Actually, because I got this involved thing, maybe we'll use a graph. Let's use a graph. Just to try all the different ways we know how. So 2, care it, and then we've got to put it in parentheses because we've got terms in our exponent. 3x minus 1, close the parentheses, and then that's going to be equal to 17. Got to go to my window and make sure that I can go to 17. And y is 17, but I'm only going up to 10 here at y max. So I need to come down to y max and change that. And I'm going to say 25 because I really don't want it to be right at the edge. 17 is pretty close to 20. So graph it, make sure I can see my intersection. And I definitely will see my intersection. And I'm going to sketch my little graph here so that I can put my work on a paper. This is y equals 17. Here's my intersection point. And the calculator tells me second trace 5, enter, enter, enter. That that is x equal 1.695. So I'm going to say that's approximately 1.7. So x, write that a little bit better, is approximately 1.7 by graph. And finally I have another log problem. And remember with log problems we convert to an exponential or we can change the base. We've already changed the base so I want to convert this one. And you start with your base and then you hop across to get the exponent. Remember logs are equal to exponents. And that's equal to hopping back across equal sign that x minus 5. Well 2 to the 5th I know is 32 or you could go to your calculator and put 2 care at 5 if you didn't know that. And so I just add 5 to both sides. And 32 plus 5 is going to be 37 and that's equal to x. And just a quick reminder whenever you want to check. You can go back to this original equation and say well 2 to the 5th is supposed to be equal to 37 minus 5. And in your calculator 2, oops, main screen, 2 care at 5 is 32. And we all know that 37 minus 5 is 32 so we can accept that x is equal to 37. Okay we have 2 more problems. Long base 5 of x is equal to 3 so we can convert. We cannot change the base. Anybody know why we can't change the base? The reason we can't change the base is because inside the parentheses is not a number. The only time we can change a base is if we have a number. All our numbers are on the log side but they're not here so we have to convert. We have to convert and then that will tell us how to solve it. So we take our 5, hop across and get this exponent of 3 and equal to x and this becomes a very simple problem. And if you don't know what 5 to the 3rd is, if you don't recognize that that's 125 you can go to your calculator and say 5 care at 3 and find out that sure enough x is equal to 125.