Chem, Half Life Question?

Question

I know two equations which I was told to use to find the half life. But I am lost as to how I have to use them. So if someone knows the answer or just how to solve this please help. Thanks.

Lortab contains gydrocodone which is a powerful pain killer. The half life of hydrocodone is about 5 hrs, so half of the drug will be out of your system in 5 hrs. Half of the half will be out in another 5 hrs.
If you take 10mg of Lortab, how many days will it take until the amount of hydrocodone in your body is down to 1 nanogram?

Answer 1

1 nanogram is 10^-9 g

Easiest way is:

1*10^-9 g= 0.010g*(1/2)^x, solve for x, tells the number of half lives needed,

10^-7 = (1/2)^x -----------------> log(10^-7) = x* log(1/2)

x = 23.253 half lives but since each half life is 5 hours it is 116.267 hours, divide that by 24 to get the number of days, which is 4.8 days.

Answer 2

The equation was probably something like a = A*2^(-t/T), where a is the amount at time t, A is the initial amount, t is the elapsed time, and T is the half-life. In this case, you've been given a = 1 ng and A = 10 mg, as well as T = 5 hr, and you're looking for t. You just need to divide both sides by A and then take the log base 2 to get -t/T equal to a number that should be negative. Then multiply both sides by T to get your final answer. Before you start, you should convert both masses to kg, or some other metric unit, as long as it's the same one.

Answer 3

10 mg = 10,000,000 ng
What power of 2 corresponds to 10,000,000?
In other words, 2^x = 10,000,000 or x = log base 2 of 10,000,000
using an obscure property of logarithms, log base 2 of 10,000,000 = log base 10 of 10,000,000 / log base 10 of 2.
log base 10 of 10 million is 7. log base 10 of 2 is .30103; log base 2 of 10,000,000 = 23.25
IT TAKES 23 1/4 HALF LIVES TO GO FROM 10 mg DOWN TO 1 ng. That's 116.25 hours, or almost 5 days.