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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.
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.
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And it'll get a little bit mathy, usually involving a little bit of algebra or a little bit of exponential decay, but to really show you how you can actually figure out the age of some volcanic rock using this technique, using a little bit of mathematics.
So we know that anything that is experiencing radioactive decay, it's experiencing exponential decay.