You are trying to determine the half-life of a new radioactive element you have isolated. you start with 1 gram, and 5 days later you determine that it has decayed down to 0.6 grams. what is its half-life? (round your answer to three significant digits.)

 The remaining mass mass of the substance at time t can be modeled with an exponential equation in this form: 

m = ab^t, 
where m is the remaining mass in grams, and t is the time in days. 

We are given the mass at time zero and at time two. 

Let t = 0. 
10 = ab^0 
a = 10 
m = 10b^y 

Let t = 2. 
7 = 10b² 
b² = 0.7 
b ≈ 0.84 
m ≈ 10(0.84)^t 

Now to find the half-life, let m = 5 g, which is half of the initial mass. Solve for t. 

5 = 10(0.84)^t 
(0.84)^t = 0.5 
t ln(0.84) = ln(0.5) 
t = ln(0.5) / ln(0.84) ≈ 3.9 days