If 250 mg of a radioactive element decays to 200 mg in 12 hours, find the half-life of the element. (Round your answer to the nearest

200 = 250 x 1/2^(12/t), where t=half-life
0.8 = 1/2^(12/t)            Take the natural log of both sides
-0.22314355 =(12/t) x -0.69314718      Cross multiply
-0.22314355 t =-8.317766                           Divide both sides by -0.22314355
t = 37.275 =~37 hours - the half-life of this material.

We have that

200 = 250(.5)^(12/h)   where h is the half-life in hours

Divide both sides by 250

200/250  = (.5)^(12/h)

.8  = (.5)^(t/h)      take the log of both sides

log (.8)  = log(.5)^(12/h)   

And by a log property we can write

log (.8) =  (12/h) log (.5)

Rearrange to isolate h

h  = 12log(.5) / log(.8)  ≈  37.28 hrs