Unit
Exponential and Logarithmic Functions
University of Michigan
Runs his own tutoring company
Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!
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University of Michigan
Runs his own tutoring company
Carl taught upper-level math in several schools and currently runs his own tutoring company. He bets that no one can beat his love for intensive outdoor activities!
So this is an example of a problem that you’ll not only see in Algebra 2 but also in your everyday life, that has to do with interest in investing. Let’s take a look.
You invest 5000 dollars at 5% interest compounded quarterly. This is going to be important so pay attention. How long will it take to double your money?
We know that our formula for dealing with interest problems is the amount is equal to the principal times 1 plus our rate, over the number of times that it’s compounded per year, all raised to the amount of time, time is that individual number. Now it seems really complicated but with word problems remember we just need to take our information, plug it into our formula and solve from there.
Let’s take the easy part first. We know that our principal is $5000 so we can plug that in right away. And we know that our rate is 5%. Now remember when we’re dealing with percent we need to convert to decimal. We can’t just write 5 that would give us a completely wrong answer, so it’s 0.05 we also know that it’s compounded quarterly so our number is 4. If it were compounded weekly it would be 52, and then this is all raised to our unknown variable because we’re trying to figure out how long it will take us so we’re looking for this t, the time. This is zero 4 times time, okay.
Now our amount, we’re trying to double our money so we start out with $5000 and we’re trying to get to $10000. Here we go, here’s our formula, we have our one unknown variable that we’re trying to solve for and now we just need to make sure to take it slow, take it step by step and to solve it. The first thing that we want to do is we want to isolate this variable so let’s get rid of this first expression, 5000. That’s pretty easy, simplify, that’s 2 is equal to 1 plus 0.05 over 4 raised to the 4t.
Now we want to isolate this variable, what we can also do is we can solve within this fraction. Let’s go up here so we have more room to work. So I did this before hand, let me just check and we know that it is 1.0125 raised to the 4t.
Here’s where it gets a little bit tricky; remember when we have a variable in the exponent, the way that we can get this variable down to solve is by taking the natural logarithm. Let’s go ahead and just take the natural log of both sides. When we do that we get the natural log of 2 is equal to, we can bring this down, 4t times the natural log of 1.0125, that was what was in our parenthesis. Now from here what we can do is we can simply, remember to take the natural log of that. We can isolate our variable.
So let’s go ahead and let’s move everything but the variable to that other side and solve. I always like to keep my variable on the left. So what we can do is we have the natural log of 2 over 4 times the natural log of 1.0125. We can go ahead and plug that into you calculator, make sure to use your parentheses and we have our final answer as approximately 13 point, let me check what I did, 13.94 years.
What this is saying is that if we invest #5000 at 5 percent interest that is compounded quarterly it will take us approximately 13.94 years to double our investment.