You have a sample of a certain radioactive substance. After two days, thirty percent of the sample has decayed. What is the half-life of the material and how much will have decayed after a week?
amount = a (1/2)^(t/k), where k is the half-life period
when t = 2, amount = .7a
.7a = a(1/2)(2/k)
.7 = (.5)^(2/k)
take log of both sides and use log rules
log .7 = (2/k)log .5
2/k = log .7 / log .5 = .514573...
k = 2/.514573.. = 3.887 days
the half-life is 3.89 days
check:
amount = 1 (1/2)^(2/3.887) = appr .7 or 70%