I'm not quite sure what you want to know about this (hence the delay in answer). So I'll just explain the variables, give some examples, then maybe some general tips.
First of all, this is a half-life formula. What that means is that, after a certain amount of time, a radioactive substance will reduce to half it's original substance. After that same amount of time again, the substance will reduce to another half. IE:
If the half-life of imaginarium is 10 days and we start out with 100 grams, then:
You begin with 100 grams of imaginarium.
Ten days later, you have 50 grams.
Ten days after that (so a total of 20 days since the start), you have 25 grams.
Ten days after that (30 total days from the start), you have 12.5 grams.
So on and so forth. Half-life formulas are a popular example of exponential decay. They graph like:
Okay. So let's define the variables in the equation you gave:
AR=AD(1/2)^(t/d)
AR is the final amount of a substance you have after a specified time.
AD is the original amount of a substance you have.
t is the time it takes to decay to the amount of AR.
d is the half life.
So let's do an example.
Let's go back to my imaginarium. Let's say we want to know how much we'll have left after 50 days. So we know:
d = 10 days
AD = 100 grams
t = 50 days
AR = ?
So we're solving for AR. The equation we have is already solved for it, so just plug-and-chug:
AR= 100(1/2)^(50/10) = 3.125 grams!
What is we want to know how much we have after 5 days? Now
t = 5 days
But the rest is still the same, so more plug-and-chug.
AR= 100(1/2)^(5/10)= 70.7 grams!
Now we want to know how long it takes to get 25 grams. (Obviously, it's 20 days because it's an easy number to work with that was already an example, but let's just go through the calculations, okay?)
Rearrange the equation for t. (If you want more help with that. If you need help with the change of base formula/rule, ask us, as I don't think that's covered in the logarithms tag!)
d * log1/2(AR/AD) = t
Plug-and-chug:
10 * log1/2(25/100) = t = 20 days!
General Notes:
Always make sure your units with your time match that if your half-life. If you want your answer in a different unit of time (days, hours, etc) use dimensional analysis on your (t/d) first. For example. If you have a t= 5 hours, but your d= 1 day, you had BETTER change your t.
Stemming from the previous bullet, it can help to think about your (t/d) as "the number of half-lives you need." t/d is a ratio; if your half-life is 10 days and you want to know the answer for 30 days, (30/10) = 3 half-lives.
Another thing that might help is noticing that when you're solving FOR t, you end up with a ratio on the left side of the equation that looks like (AR/AD). That's kind of like the percent (in decimal form) of the original you want to find. So if you want how long it takes for 100 grams of something to turn into 30 grams of something, you want to know how long it takes for you sample to be 0.3, or 30% the size of the original.
You'll never get an answer of 0. You'll be close, but you'll never actually reach it because you're always halving.
SURPRISE, SURPRISE, imaginarium is not a real element. Sorry :)
To be perfectly honest, this is the basic rundown of half-life. The last time I learned half-life was in my college chem class, and BOY was that a different kettle of radioactive fish. (Your PSA of the day: Don't eat radioactive fish!) That had to do with what order of reaction the equation was in, and something about rate laws and concentrations and- yeah. Anyway, if you were looking for a answer at that level of chemistry, specify that in another ask with preferably more details about exactly what's bugging you (maybe a problem?), and someone who actually knows how to do that stuff will answer. (...I'm still slightly bitter I missed those questions on the test last semester. DARN YOU, GEN CHEM II, DARN YOU TO HECK!)
--Lolo