 This video will be solving equations again, but this time we're going to be using applications. So we have a number of Franklin's goals in North America has been in significant decline in the past years. Population of Franklin's goals can be modeled by this function f of t is equal to 750 times 0.9405 to the t. Notice that base is less than one, so that tells us that it's declining. And we're going to let f of t represent the number of Franklin's goals in thousands. And then in North America, and t is since 2000. So when we want to estimate the goal population in 2005, we know that t is equal to 5, 5 years after 2000. And we have the function, we're really doing f of 5 and that's equal to 750 times that 0.9405 raised to the fifth. So go to our calculator, 750 parenthesis 0.9405 and close that parenthesis, carry it 5. And we find out that there are approximately 551.9 and this is in thousands. Okay, so when will the Franklin's goals reach 500,000? Now remember, it's in thousands. So we really, we want to know this but when we write the equation we really only need the 500 because the thousand knowing our problem is in thousand takes care of those three zeros for us. So we really have 500 is equal to that 750 times 0.9405 to the t. This time we're solving for the exponent. So remember, when you have an exponential equation with a variable as you're unknown, you've got to isolate that base in the exponent. This is 750 times that so I'm going to divide both sides by 750. And remember, we want to have as exact an answer as possible so we're going to leave it as a fraction. 500 over 750 is going to be equal to our 0.9405 to the t. So we have an exponential here that we can just take the log of both sides. You know me, I tend to take logs instead of natural logs but if you want to take a natural log you could say ln instead of log everywhere that I'm writing log. Equal to log of 0.9405 to the t and we rewrite this. The side stays the same because it's just a number. Log of 500 over 750 is going to be equal to and we bring the exponent down front in front of that log. So t times the log of 0.9405 and we divide the log and we divide the log. Now we know that t is equal to this log and I'm not going to rewrite it. I'll just write my answer like this. That is exact. If we wanted to know approximately what that is, we'd have log 500 divided by 750 and maybe we asked you to write it as a decimal. Divided by the log of 0.9405, close the parenthesis. So in approximately, if we asked you how many years you'd need to go to the decimal. So in approximately 6.6 years or 7 years if you want to say after 2000, there will be 2000. There will be 500,000 goals left. And then just your nice environmental question. What might be causing this population to stop declining at this rate? What could we do to make it stop? We really don't know what's happening but that might be how it's plastic in landfills. Maybe if we quit doing that or maybe if the weather is causing them or their food, amount of food that they have, just being able to fix those kinds of things might make it better. Alright, we also have interest problems that we can do that work wonderfully with trying to solve these equations. So a person wants to invest $50,000 at 6% interest but cannot decide if they should compound annually or continuously. So let's look at both. Remember that the equations are a is equal to p times 1 plus r over n to the nt. But if n is 1 then it's 1 plus r. And a is equal to p times e to the rt for continuous. How much would they make if they invested annually? Well, we have a being equal to 50,000 that they invested. That's the p times 1 and our rate or n is 1. So we're going to just say 1 plus the rate 0.06 because we just would be dividing by 1 to the nt. But it's really just a 20 because it's 20 times 1. And when we take that to our calculator we get 50,000 per 1.06 when we add those two. Carat 20. So in 20 years they're going to have $1,60356.77 after 20 years at 6% annual. If we wanted to know continuously we would have a equal to our 50,000. That's our p. And then e is our base to the r which is 0.06 times the t which is 20 years. So this time we have more in our exponent than 50,000. And then second ln remember we'll get you to the e and then we can write 0.06 times 20 and then close the parenthesis. And we get approximately 1,66005.85 if we round it. So $166,005.85 after 20 years at 6% continuous interest. And we have how long would it take to double their money if the same investment was compounded continuously at 6%. So doubling their money they started out with 50,000. So if we double that that's going to give us 100,000 that we know is in the account. So that's the a. So 100,000 is equal to the p, 50,000 times e to the r which is 0.06 but we don't know how long so it's t. So there's our equation and we want to get to this base with this exponent all by itself. That's always where we're going when we have an exponential. So we divide by 50,000 and 100,000 divided by going to be 2 is equal to e to the 0.06t. Since this is base e then natural log is the way to go. Nice things happen when you take the natural log although you could take the log of both sides. But if we take the natural log then we have the natural log of 2 on this side is equal to the natural log of e to the 0.06t. And the property says bring the exponent out front so we still have ln 2 on this side. And that's equal to our exponent of 0.06t and then times that ln of just plain e. Come over to your calculator for a second. ln e to the first is equal to, because that's what we have. ln and then second ln and the exponent is 1. You have to close that parenthesis twice because it's one for the exponent and one for the argument. And we find that that's just equal to 1. So we really can just say that this is 0.06t times 1. That's why taking the natural log is so nice because at this point then we could just divide by 0.06 and divide by 0.06. And t will be equal to ln 2 divided by 0.06. Remember base e take the natural log of both sides and then ln e will always be equal to 1. Finally, how long will it take to double our money if it's annual? Again, we know it's 100,000. We've been working with that. And it's equal to 50,000 times our 1.06 since it's annual to the t. Divide off the 50,000 again because of course we have to get this base with this exponent all by itself. So 1.06 to the t is going to be equal to 2. And I want to graph this one since we did the natural log of both sides for the other one. And if we come up here we have 2 and on the other side we have 1.06kx. Second trace, I think my window is okay. We'll find out. Here's my 2 and here comes my exponential and we're good. Actually my graph because I changed it earlier is goes to 15 and this goes to 10. So here's my 2 and here's my exponential coming through here. This is y equal to and x is going to be equal to enter, enter, enter 11.895 so 11.9 years. Which method of compounding would you choose and why? We remember when it was annual we could double our money and we just found out 11.8 or 9 years, almost 12 years. And when we did it by, we never did figure out what this number was when we did it natural log or when we had base e continuous. Well in 2 divided by .06 gives us 11.55 so continuous was 11.6 so I would go with continuous because it doubles faster. So you're going to get more money quicker if you go continuously.