How to rearrange a radioactive decay equation y = mx +c

Question

I have the equation $\frac{dN}{dt}= - Nk$ where $k$ is the decay constant.

When $time = 0$,

we get $N(t) = N(0) e^{-kt}$.

How would I rearrange this to the $y = mx + c$ format? How would I find the decay constant? Thanks in advance.

Answer 1

You can transform the equation into a linear equation, but then you have to take logs of $N(t)$ and $N(0)$

$\ln(N(t))=\ln \left[(N(0))\cdot e^{k\cdot t}\right]$

$\ln(N(t))=\ln (N(0))+\ln\left[ e^{k\cdot t}\right]$

$\ln(N(t))=\ln (N(0))+k\cdot t$

This is equivalent to $y=c+mx$

Let´s say the origin values are

t      0  1  2  3  4

N(t)  16  8  4  2  1

Now you calculate the logs of $N(t)$. The values of $t$ musn´t be transformed. Two value pairs are sufficient to evaluate the value of $k$.

Remark

I have $+k$ at the function, not $-k$. But it doesn´t matter. If you have a decay then the value of k is negative.