SOLUTION: carl plans to invest $500 at 8.25% INTREST, COMPOUNDED CONTINUOUSLY. how long will it take for his money to triple? use y= Pe^rt. show your work

Algebra ->  Logarithm Solvers, Trainers and Word Problems -> SOLUTION: carl plans to invest $500 at 8.25% INTREST, COMPOUNDED CONTINUOUSLY. how long will it take for his money to triple? use y= Pe^rt. show your work      Log On


   

Question 242151: carl plans to invest $500 at 8.25% INTREST, COMPOUNDED CONTINUOUSLY. how long will it take for his money to triple? use y= Pe^rt. show your work
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
After tripling $500 we will have $1500. So the equation we have to solve is:
1500+=+500e%5E%280.0825t%29

First we'll divide both sides by 500:
3+=+e%5E%280.0825t%29
Now, since the variable is up in the exponent, we will use logarithms to solve for it. The natural choice (excuse the pun) is to use natural logarithms (aka ln) because e is the base:
ln%283%29+=+ln%28e%5E%280.0825t%29%29
Using the property of logarithms, log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29, we can move the exponent in the argument out in front as a coefficient:
ln%283%29+=+%280.0825t%29ln%28e%29
Since ln%28e%29+=+1 by definition we now have:
ln%283%29+=+%280.0825t%29
Divide both sides by 0.0825:
ln%283%29%2F0.0825+=+t
This is an exact answer. You can use your calculator to find a decimal approximation for t (which is the time it will take to triple your money).

If your calculator doesn't "do" ln, we can use base 10 logs instead of ln:
log%28%283%29%29+=+log%28%28e%5E%280.0825t%29%29%29
log%28%283%29%29+=+%280.0825t%29log%28%28e%29%29
Since we used base 10 logs, log(e) will not "disappear" like ln(e). So we are "stuck" with it. Divide both sides by 0.0825log(e):
log%28%283%29%29%2F%280.0825%2Alog%28%28e%29%29%29+=+t (e is approximately equal to 2.7182818284590451. Round this off as you choose and then use your calculator to find t.

Although log%28%283%29%29%2F%280.0825%2Alog%28%28e%29%29%29 and ln%283%29%2F0.0825 do not look the same they are actually equal.