# Thread: Decay rates in exponential decay

1. ## Decay rates in exponential decay

Hi there, I'm actually trying to write a little computer program which will invlove me using a half-life type of decay, and I'm not really sure how to go about various calculations.

What I want to calculate is a particular value, x, to repeatedly multiply my decaying variable Alpha by (Alpha is given and can be between 0.01 and 1), so that over 1000 steps Alpha becomes 0.01.

Now, I did this by hand for an initial alpha of 1 and got a value 0.995405418.

I'm thinking that could possibly boil down to

0.01 = Alpha ( x ^ 1000)

But I don't have the math skills to rearrange for x, and I'm not sure that's the right answer anyway.

Any pointers?

2. Originally Posted by misterprogrammer
Hi there, I'm actually trying to write a little computer program which will invlove me using a half-life type of decay, and I'm not really sure how to go about various calculations.

What I want to calculate is a particular value, x, to repeatedly multiply my decaying variable Alpha by (Alpha is given and can be between 0.01 and 1), so that over 1000 steps Alpha becomes 0.01.

Now, I did this by hand for an initial alpha of 1 and got a value 0.995405418.

I'm thinking that could possibly boil down to

0.01 = Alpha ( x ^ 1000)

But I don't have the math skills to rearrange for x, and I'm not sure that's the right answer anyway.

Any pointers?
the way you have set up your equation, $\alpha$ does not become $0.01$, rather the product of $\alpha$ and $x^{1000}$ becomes $0.01$

it sounds like you want $\alpha$ to be a given fixed constant $0.01 \le \alpha \le 1$

to solve for $x$ ...

$0.01 = \alpha \cdot x^{1000}$

$\frac{0.01}{\alpha} = x^{1000}$

$\left(\frac{0.01}{\alpha}\right)^{\frac{1}{1000}} = x$

so ... say the given $\alpha = 0.5$

$\left(\frac{0.01}{0.5}\right)^{\frac{1}{1000}} = x$

$(0.02)^{\frac{1}{1000}} = x$

$x = 0.996095619...$

3. Hello, misterprogrammer!

I hope I've interpreted your intentions correctly.

I'm trying to write a computer program which invloves a half-life type of decay.

What I want to calculate is a particular value, $x$ to repeatedly multiply my initial amount $A_o$
so that over 1000 steps $A_o$ is reduced by a factor of 0.01

Now, I did this by hand for an initial alpha of 1 and got a value 0.995405418.
. . This is correct!

We have: . $A \:=\:A_ox^t$

When $t = 1000$, we want $A$ to be 0.01 of $A_o$

So, we have: . $0.01A_o \:=\:A_ox^{1000} \quad\Rightarrow\quad x^{1000} \:=\:0.01$

Therefore: . $x \;=\;(0.01)^{\frac{1}{1000}} \;=\;0.995405417$