SOLUTION: Please help!!! The half life of an element is 3.9 * 10^9 yr. How long does it take a sample of the element to decay to 2/5 of its original mass? Use A=A0(1/2)^(t/T) where AO is

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Question 576524: Please help!!!
The half life of an element is 3.9 * 10^9 yr. How long does it take a sample of the element to decay to 2/5 of its original mass? Use A=A0(1/2)^(t/T) where AO is the initial amount, T is the half life, and t is the time. (express results in scientific notation, rounded to the nearest hundredth)
Answer by KMST(5397) About Me  (Show Source):
You can put this solution on YOUR website!
A=A%5B0%5D%281%2F2%29%5E%28t%2FT%29 is the equation given and you want the value of t that will make A%2FA%5B0%5D=2%2F5
(You know it has to be more than T=3.9+%2A+10%5E9years because at T you would have 1/2 left and 2/5<1/2).
We could divide both sides by A%5B0%5D to get
A%2FA%5B0%5D=%281%2F2%29%5E%28t%2FT%29
Taking logarithm of both sides
log%28%28A%2FA%5B0%5D%29%29=%28t%2FT%29%2Alog%28%281%2F2%29%29 --> log%28%28A%2FA%5B0%5D%29%29=%28t%2FT%29%2A%28log%281%29-log%282%29%29 --> log%28%28A%2FA%5B0%5D%29%29=%28t%2FT%29%2A%280-log%282%29%29 --> log%28%28A%2FA%5B0%5D%29%29=%28t%2FT%29%2A%28-log%282%29%29 --> log%28%28A%2FA%5B0%5D%29%29=-%28t%2FT%29%2Alog%282%29
Plugging in the values given,
log%28%282%2F5%29%29=-%28t%2F%283.9+%2A+10%5E9%29%29%2Alog%282%29
Using the approximate values for the logarithms
-0.39794=-0.30103%2At%2F%283.9+%2A+10%5E9%29
So, t=0.39794%2A%283.9%2A+10%5E9%29%2F0.30103
highlight%28t=5.16%2A10%5E9years%29