# Thread: Half-life problem, not sure how to figure out "t" from this formula

1. ## Half-life problem, not sure how to figure out "t" from this formula

I'm stuck on a half-life problem.
The formula is A=A0(1/2)t/k where k is the half-life. I am given the half-life and A/A0 but I have no idea how I'm supposed to go from that to finding t. The book does not give me an example, the homework website just leaps instantly from that formula to the answer for a sample problem. I have been using natural log as we were using it earlier to do other half-life problems but the natural log doesn't give me any help so obviously I'm not clear on the steps. It's most likely something so basic that I cannot remember it.

2. ## Re: Half-life problem, not sure how to figure out "t" from this formula

Divide both sides by $\displaystyle A_0$ to get $\displaystyle \frac{A}{A_0}= (1/2)^{t/k}$. You say you are told the "half life" which is the value of t so that $\displaystyle (1/2)^{t/k}= A/A_0= 1/2$. Of course, $\displaystyle 1/2= (1/2)^1$ so T/k= 1, k= T where "T" is that "half life". $\displaystyle \frac{A}{A_0}= (1/2)^{t/T}$

Now, what "t" are you trying to find? If just "t in terms of A", taking the logarithm of both sides, $\displaystyle (t/T)log(1/2)= log(A/A_0)$. Can you finish it now?

3. ## Re: Half-life problem, not sure how to figure out "t" from this formula

I was confused by the answer given above. I would just divide both sides by A sub 0 and then take log base 1/2 of both sides... if you don't like that base use base formula (log base 10 A/a)/(log base 10 1/2). Multiply both sides by k and now you just pop it in the old calculator.

Edit: I'm really confused when it comes to formatting so hopefully you can understand it written out.

4. ## Re: Half-life problem, not sure how to figure out "t" from this formula

In general, for a and b any two numbers, $\displaystyle log_b(x)= \frac{log_a(x)}{log_a(b)}$.

That's useful for shifting logarithms from one base to another. Do you know what logarithms are?