So let's figure out the general formula for carbon.

## Illustrative Mathematics

Carbon, that's the one that we addressed in the half-life. We saw that carbon has a half-life of 5, years. So let's see if we can somehow take this information and apply it to this equation. So this tells us that after one half-life-- so t is equal to 5, N of 5, is equal to the amount we start off with. So we're starting off with, well, we're starting off with N sub 0 times e to the minus-- wherever you see the t you put the minus 5, so minus k, times 5, That's how many years have gone by. So if we try to solve this equation for k, what do we get? Divide both sides by N naught. If we take the natural log of both sides, what do we get?

Exponential Equations: Half-Life Applications

The natural log of e to anything, the natural log of e to the a is just a. I just took the natural log of both sides. The natural log and natural log of both sides of that. But let's see if we can do that again here, to avoid-- for those who might have skipped it. So it equals 1. So now we have the general formula for carbon, given its half-life. At any given point in time, after our starting point-- so this is for, let's call this for carbon, for c the amount of carbon we're going to have left is going to be the amount that we started with times e to the minus k.

This is our formula for carbon, for carbon If we were doing this for some other element, we would use that element's half-life to figure out how much we're going to have at any given period of time to figure out the k value. 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. 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.

## Carbon 14 dating

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.

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