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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. And let's say that the argon-- actually, I'm going to say the potassium-40 found, and let's say the argon-40 found-- let's say it is 0.01 milligram. And to figure out our initial amount, we just have to remember that for every argon-40 we see, that must have decayed from-- when you have potassium-40, when it decays, 11% decays into argon-40 and the rest-- 89%-- decays into calcium-40. So however much argon-40, that is 11% of the decay product.So how can we use this information-- in what we just figured out here, which is derived from the half-life-- to figure out how old this sample right over here? So we need to figure out what our initial amount is. So if you want to think about the total number of potassium-40s that have decayed since this was kind of stuck in the lava.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.
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.Or I could write it as negative 1.25-- let me write times-- 10 to the ninth k. Or you could view it as multiplying the numerator and the denominator by a negative so that a negative shows up at the top. The negative natural log-- well, I could just write it this way.And so we could make this as over 1.25 times 10 to the ninth. If I have a natural log of b-- we know from our logarithm properties, this is the same thing as the natural log of b to the a power.Archaeologists use the exponential, radioactive decay of carbon 14 to estimate the death dates of organic material.The stable form of carbon is carbon 12 and the radioactive isotope carbon 14 decays over time into nitrogen 14 and other particles.