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# After how many half-lives will (a) 10.0%, (b) 5.00%, and (c) 1.00% of a radioactive sample remain?

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in number 61 were asking, How many half life's does it take until you're down to 10% of your original sample? 5% of it and 1% of it so amusing. This equation we're the remaining amount of radioactive is the original times 1/2 raised to the number of half lifes. And that's what I'm looking for us this end. So my original MT. My new amount is 10% of my original mount, so I can think of that as 100.10 times the original amount. Here's what I know mountainous. So you can tell that cancels and also solved this. I'm going to take the log of both sides so ever here I have a log of 0.10 over here I have a little log of 0.5 raised to the end, But remember how logs work instead of for that exponents, I bring that out in front. So this is and so sell for m I would just divide So over here, man, have a log of quite 10 divided by log 0.5 and that we go and and and is the number half lifes. So I do that I get 3.32 That's money. Half life's until about 10% We can sell over here. I mean, we're gonna end up doing the same thing over and over again. So for here, I'm just gonna skip to the bottom line here because the only thing that's gonna change is this percent it's just being 10% number 85%. So that's going to change this. So my bottom row here is gonna be a log of 5%. So 50.5 divide by log a 0.5 and they get 4.32 It's money. Half life's until mid five person and then for 1%. Now, instead of having 10% we're gonna have 1%. So log, appoint no one to every log 0.5. That's gonna be my my end. And it is six 0.64 So after 6.6 4/2 lives, I would be down to 1%

University of Virginia