First setup the equation for how much element is remaining after each day:
y = a ( 100% - 3.418%)^t
where y is the amount remainig and t is the number of days
We can clean this up by subtracting the percentages and converting that to a decimal:
y = a (0.96582)^t
Now for the half-life, we need to figure out how long until there's half of what we started with. When does "a (0.96582)^t" equal "1/2 of a"
1/2a = a (0.96582)^t
divide by a on both sides and you get
1/2 = (0.96582)^t
take the log of both sides and you get:
log(1/2) = t log(0.96582)
divide both sides by log(0.96582)
log(1/2) / log(0.96582) = t
Throw that into a calculator and you have
t ≈ 19.93
