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remaining, after time
is a fraction of the initial amount
being the half-life.
(Note: There are many (equivalent) ways to express that relationship/function, but this is the one that seems to make more direct sense: the fraction is
multiplied times itself as many times as half-lives have passed).
So, taking logarithms of both sides
For the problem, with 33% remaining,
So, we can find
(in days), by solving
So the half-life is 5 days.