SOLUTION: Please help me solve this word problem: A certain radioactive element decays at a rate of 4% per year. Find the half-life of the substance (i.e. the time it will take for one half

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Question 320172: Please help me solve this word problem: A certain radioactive element decays at a rate of 4% per year. Find the half-life of the substance (i.e. the time it will take for one half of any given amount of the substance to decay).
Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
Please help me solve this word problem: A certain radioactive element decays at a rate of 4% per year. Find the half-life of the substance (i.e. the time it will take for one half of any given amount of the substance to decay). 

Let A be the original amount

After 1 year only 96% will remain, which will be 0.96A
After 2 years only 96% of 0.96A will remain, which is 0.96%5E2%2AA
After 3 years only 96% of 0.96%5E2%2AA will remain, which is 0.96%5E3%2AA
...
After n years only 96% of 0.96%5E%28n-1%29A will remain, which is 0.96%5En%2AA

We want to know how many years before what will remain will be only 1%2F2A or 0.5A 

0.96%5En%2AA%22=%220.5A

Divide both sides by A

0.96%5En%22=%220.5

Take logs of both sides:

log%28%280.96%5En%29%29%22=%22log%28%280.5%29%29

Use the rule of logs that says log%28B%2C%28A%5EC%29%29=C%2Alog%28B%2C%28A%29%29 to bring
the exponent n in front of the log

n%2Alog%28%280.96%29%29%22=%22log%28%280.5%29%29

Divide both sides by log%28%280.96%29%29

n%22=%22log%28%280.5%29%29%2Flog%28%280.96%29%29%22=%2216.97974802

or 17 years, rounded to the first whole year that the amount will be
no more than half the original amount.

Edwin