Use applications of exponential functions and logarithmic functions to solve.

Question

The half-life of plutonium 241 is 14.4 years.If 100 grams is present now, How much will it take to reach 2 grams of plutonium 241?
y=a(1/2)^(t/14.4)
How do I solve for t? I'm not sure where to even begin.

Answer 1

100(1/2)^(t/14.4) = 2
(1/2)^(t/14.4) = 0.02
t/14.4 ln(1/2) = ln(0.02)
t/14.4 = ln(0.02)/ln(0.5)
That is just log_0.50.02
t = 14.4 ln(0.02)/ln(0.5) = 81.27

That's about 5.6 half-lives. That makes sense, since
(1/2)^5 = 1/32 and
(1/2)^6 = 1/64
Your fraction (1/50) is between those two values.