# Thread: How to use exponential function?

1. ## How to use exponential function?

For this word problem: An isotope of sodium has a half-life of 20 hours. Suppose an initial sample of this isotope has mass 10 grams.The amount of the isotope (in grams) remaining after t hours using the exponential decay function would look like what?

I'm not sure what it would look like. I know it's in the form of: amount(another amount)^t but I'm not sure how to put the amounts in for this work problem. Thanks!

2. ## Re: How to use exponential function?

What book are you using? It sounds like you need to read the chapter that covers this. It is harder for us to explain, and probably just as much reading, as it would be for you to read it for yourself in your textbook. Why are you asking without even attempting it?

3. ## Re: How to use exponential function?

Originally Posted by SlipEternal
What book are you using? It sounds like you need to read the chapter that covers this. It is harder for us to explain, and probably just as much reading, as it would be for you to read it for yourself in your textbook. Why are you asking without even attempting it?
I have attempted it, and we have gone over the lesson, but I don't think I'm doing it right, because I keep getting different answers. I've gotten both of these: 20(1.071773462)^t 20(0.9330329915)^t and One of my friends in the same class disagrees with me and is saying it's supposed to be a 10 where I have the 20.

4. ## Re: How to use exponential function?

Originally Posted by SlipEternal
What book are you using? It sounds like you need to read the chapter that covers this. It is harder for us to explain, and probably just as much reading, as it would be for you to read it for yourself in your textbook. Why are you asking without even attempting it?
I think I figured it out, would it be 10(.965936329)^t? Because if t is then 20 then it equals 5. Is this right?

5. ## Re: How to use exponential function?

As slipEternal suggested, it would be very helpful for you to give yourself a crash course on exponential functions. Let me put it this way:

f(x) = ax + b defines a linear function.
f(x) = ax^2 + bx + c (a is not zero) defines a quadratic equation.

Etc. What defines an exponential function?

6. ## Re: How to use exponential function?

In general, half-life problems use this equation:

Let $f(t)$ be a function that gives the amount of isotope remaining after time $t$ has passed. Then $f(t) = M_0 e^{-kt}$ where $M_0$ is the initial amount of isotope and $k$ is a positive constant. So, plugging in the initial mass, you have $f(t) = 10e^{-kt}$ where $t$ is in hours and $f(t)$ gives the amount of remaining isotope in grams.

Then you are told the half-life is 20 hours. So, when $t=20$, $f(t) = \dfrac{10}{2} = 5$. So, let's look at that equation:

$5 = 10e^{-k(20)}$

We want to solve for $k$. Dividing both sides by $10$ and taking the natural log of both sides gives:

$\ln \left(\dfrac{1}{2} \right) = -20k$

Dividing both sides by -20 gives:

$k = -\dfrac{1}{20}\ln \left(\dfrac{1}{2} \right)$

Now, you can use some rules of logarithms to simplify this:

$k = \dfrac{1}{20}\ln 2$

Next, use some rules of exponents:

\begin{align*}f(t) & = 10e^{-kt} \\ & = 10\left(e^{\ln(2)}\right)^{-t/20} \\ & = 10\cdot 2^{-t/20}\end{align*}