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The natural log is just saying-- to what power do I have to raise e to get e to the negative k times 1.25 billion? k is equal to the natural log of 1/2 times negative 1.25 times 10 to the ninth power.So the natural log of this-- the power they'd have to raise e to to get to e to the negative k times 1.25 billion-- is just negative k times 1.25 billion. And then, to solve for k, we can divide both sides by negative 1.25 billion. And what we can do is we can multiply the negative times the top.Accdez, par exemple, au vocabulaire juridique employ par les plus grands cabinets d'avocats dans de nombreuses langues.Many of the settlers learned to grow certain crops without any watering at all.And we know that there's a generalized way to describe that.And we go into more depth and kind of prove it in other Khan Academy videos.And now, we can get our calculator out and just solve for what this time is. So this is 1 divided by 1 plus 0.01 divided by 0.11. And it's going to be in years because that's how we figured out this constant.
So we could actually generalize this if we were talking about some other radioactive substance.So we got the natural log of 1 over 1 plus 0.01 over 0.11 over negative k. We're just dividing both sides of this equation by negative k. So let's take the natural log of our previous answer. If you saw a sample that had this ratio of argon-40 to potassium-40, you would actually be able to do that high school mathematics.Negative k is the negative of this over the negative natural log of 2 over 1.25 times 10 to the ninth. You would be able to do that to figure out this is a 157-million-year-old sample of volcanic rock.How do we figure out how old this sample is right over there? And we learned that anything that was there before, any argon-40 that was there before would have been able to get out of the liquid lava before it froze or before it hardened. Let's see how many-- this is thousands, so it's 3,000-- so we get 156 million or 156.9 million years if we round.Well, what we need to figure out-- we know that n, the amount we were left with, is this thing right over here. And that's going to be equal to some initial amount-- when we use both of this information to figure that initial amount out-- times e to the negative kt. So to figure out how much potassium-40 this is derived from, we just divide it by 11%. And this isn't the exact number, but it'll get the general idea. So this is approximately a 157-million-year-old sample.
So maybe I could say k initial-- the potassium-40 initial-- is going to be equal to the amount of potassium 40 we have today-- 1 milligram-- plus the amount of potassium-40 we needed to get this amount of argon-40. And that number of milligrams there, it's really just 11% of the original potassium-40 that it had to come from. And so our initial-- which is really this thing right over here. This is going to be equal to-- and I won't do any of the math-- so we have 1 milligram we have left is equal to 1 milligram-- which is what we found-- plus 0.01 milligram over 0.11. And what you see here is, when we want to solve for t-- assuming we know k, and we do know k now-- that really, the absolute amount doesn't matter. Because if we're solving for t, you want to divide both sides of this equation by this quantity right over here. We're going to divide that by the negative-- I'll use parentheses carefully-- the negative natural log of 2-- that's that there-- divided by 1.25 times 10 to the ninth. So the whole point of this-- I know the math was a little bit involved, but it's something that you would actually see in a pre-calculus class or an algebra 2 class when you're studying exponential growth and decay.