SOLUTION: how do i find the growth rate in this exponential growth function: 1,000,000=10,000(1+r)^27 ?

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Question 27908: how do i find the growth rate in this exponential growth function: 1,000,000=10,000(1+r)^27 ?
Answer by bmauger(101) About Me  (Show Source):
You can put this solution on YOUR website!
1000000=10000%281%2Br%29%5E27
Here's how I like to solve these types of problems (there's more than one way).
First, divide both sides by 10,000 to get:
100=%281%2Br%29%5E27
Now take the log base 10 of each side (I'd usually use the natural log-ln, but since log(100) comes out even, might as well use it)
2=log%2810%2C%281%2Br%29%5E27%29
From laws of exponents and logs recall that:
log%28b%2CM%5EN%29=N%2Alog%28b%2CM%29
Use that to write:
2=27%2Alog%2810%2C+%281%2Br%29%29
Divide to get:
2%2F27=log%2810%2C+%281%2Br%29%29
Take both sides and set them as exponent to get:
10%5E%282%2F27%29=10%5E%28log%2810%2C%281%2Br%29%29%29
Again, recall a log/exponent rule:
b%5Elog%28b%2CM%29=M
Use that to write:
10%5E%282%2F27%29=1%2Br
10%5E%282%2F27%29-1=r
Which evaluates to approximately 18.597%
Double checking:
10000%2A%281.18597%29%5E27=999997Check.
Another way to do this problem (if your calculator will evaluate it and teacher will allow) is to take the reciprical exponent of both sides.
100=%281%2Br%29%5E27So we'll take both sides and raise it to the 1/27 power.
100%5E%281%2F27%29=%28%281%2Br%29%5E27%29%5E%281%2F27%29From the exponent rules, you multiply the powers on the right and 27*(1/27) gives you 1 and drops out, leaving:
100%5E%281%2F27%29=1%2Br
This is the equivilent of
root%2827%2C100%29=1%2Br
root%2827%2C100%29-1=r If your calculator will evaluate 27th roots, you will get the same answer of 18.597%