# Carbon dating and logarithms

### Exponential Decay - Math Central

Archaeologists use the exponential, radioactive decay of carbon 14 to estimate age of the fossil (or the date of death) and ln() is the natural logarithm function. where ln is the natural logarithm, Nf/No is the percent of carbon in the sample compared to the amount in living tissue, and t1/2 is the half-life of carbon The task requires the student to use logarithms to solve an exponential equation in the realistic context of carbon dating, important in archaeology and geology.

In the case of radiocarbon dating, the half-life of carbon 14 is 5, years. This half life is a relatively small number, which means that carbon 14 dating is not particularly helpful for very recent deaths and deaths more than 50, years ago. After 5, years, the amount of carbon 14 left in the body is half of the original amount. If the amount of carbon 14 is halved every 5, years, it will not take very long to reach an amount that is too small to analyze. When finding the age of an organic organism we need to consider the half-life of carbon 14 as well as the rate of decay, which is —0.

How old is the fossil? We can use a formula for carbon 14 dating to find the answer. So, the fossil is 8, years old, meaning the living organism died 8, years ago. Plutonium Plutonium is a man-made radioactive isotope. Plutonium is used to make nuclear explosives. Plutonium has a half-life of 24, years, which means that it would takeyears to decay to a safe amount. Plutonium decays exponentially into lead, but it causes concerns for humans because the tiny particles of plutonium react with oxygen and water and can be extremely flammable.

Since the half-life of Plutonium is so high even in comparison to the carbon 14 half-life of 5, years humans must be very cautious of the way they dispose of plutonium. Scientists are looking for safe ways for disposing plutonium. I am just learning the recipe so it takes me more time to look back and forth and double check. The more cookies I make, the more practice I have and the less time it takes me to bake the cookies. So let's use this to solve a problem.

Let's say that I start off with, I don't know, say I start off with grams of carbon, carbon And I want to know, how much do I have after, I don't know, after years? How much do I have? Well I just plug into the formula.

## K-Ar dating calculation

N of is equal to the amount that I started off with, grams, times e to the minus 1. So what is that? So I already have that 1. So let me say, times equals-- and of course, this throws a negative out there, so let me put the negative number out there. So there's a negative. And I have to raise e to this power. So this is equal to N of The amount of the substance I can expect after years is equal to times e to the minus 0.

And let's see, my calculator doesn't have an e to the power, so Let me just take e. I need to get a better calculator. I should get my scientific calculator back. But e is, let's say 2.

So this is equal to grams. So just like that, using this exponential decay formula, I was able to figure out how much of the carbon I have after kind of an unusual period of time, a non-half-life period of time. Let's do another one like this. Let's go the other way around. Let's say, I'm trying to figure out. Let's say I start off with grams of c And I want to know how long-- so I want to know a certain amount of time-- does it take for me to get to grams of c?

So, you just say that grams is how much I'm ending up with. It's equal to the amount that I started off with, grams, times e to the minus k.

### K-Ar dating calculation (video) | Khan Academy

And now we solve for time. How do we do that? Well we could divide both sides by What's divided by ? So you get 0. You take the natural log of both sides. You get the natural log of 0.

And so t is equal to this divided by 1. So the natural log, 0.

## Carbon 14 dating

So let me do the math. So if you have 0. This is all a negative number.