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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.

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.

So you get this side-- the left-hand side-- divide both sides. So it's negative natural log of 2 divided by 1.25. But the whole point I wanted to do this is to show you that it's not some crazy voodoo here.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.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.

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