SOLUTION: using this function, during what year was the united states total federal debt 500 billion dollars? y=1.21(1.08)^x y= debt

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: using this function, during what year was the united states total federal debt 500 billion dollars? y=1.21(1.08)^x y= debt      Log On


   

Question 393762: using this function, during what year was the united states total federal debt 500 billion dollars?
y=1.21(1.08)^x
y= debt
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
Since y is the debt and we are interested when the debt is 500 billion we will replace the y with 500 billion:
500000000000+=+1.21%281.08%29%5Ex

To solve an equation where the variable is in an exponent you usually isolate the base and its exponent, find the logarithm of each side and then solve the resulting equation. To isolate the base and its exponent we divide both sides by 1.21:
500000000000%2F1.21+=+%281.08%29%5Ex
Next we find the logarithm of each side. The base of logarithm does not make a big difference. But if you want a decimal approximation for the answer it would be wise to choose a logarithm whose base your calculator "knows". lie base 10 or base e (aka ln). I'll use base 10:
log%28%28500000000000%2F1.21%29%29+=+log%28%28%281.08%29%5Ex%29%29
Next we use a property of logarithms, log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29, to move the exponent out in front as a coefficient. It is this very property that is the reason we use logarithms on problems like this. The property allows us to move the exponent (where the variable is) to a place where we can then solve for the variable.
log%28%28500000000000%2F1.21%29%29+=+x%2Alog%28%281.08%29%29
To solve for x all we need to do now is to divide both sides by log(1.08):
log%28%28500000000000%2F1.21%29%29%2Flog%28%281.08%29%29+=+x
This is an exact expression for the answer. For a decimal approximation we get out our calculators:
log%28%28413223140495.8677686%29%29%2Flog%28%281.08%29%29+=+x
11.6161846340195687%2F0.0334237554869497+=+x
347.5427720429293512 = x
Since the United States did not exist in the year 347 this answer does not make sense. There are two reasons which may explain this answer:
  • The debt in the equation was meant to be expressed in billions of dollars. If so then we should have used 500 and not 500000000000 for y. In this case our answer works out to be 78.2732100538809163 = x. This still doesn't make sense.either. However, ...
  • These kinds of equations often use not the year itself but a number of years since some starting point year. You did not mention this in your post. But I think this is the best explanation for the answer we got. If "x" is the number of years since some starting point year, then we should add x to the starting point year to get the year when the debt was 500 billion.

My guess (and it is just a guess) is that we were supposed to use 500 and not 500000000000 for y and that the starting point year was 1900. This would make the answer to the problem: 1978.

In the future please post every detail of a problem.