It's a differential equation.
I always used 'y' instead of n so I'm going to do that for this post. Also the decay constant is generally 'k'.
Let 'change of y' = dy.
Let 'change of t' = dt.
Let 'decay constant' = k.
We have dy/dt = -ky (change of n/change of t = -decay constant * n)
This means that the rate of change of a population 'y' over a period of time 't' is proportional to the current population (ky). The negative sign means the rate is decreasing. This makes sense: over time the population will decrease.
To solve it you have to do some quick math:
dy = -ky * dt (multiply both sides by dt)
dy/y = -k * dt (divide both sides by y)
Now this can be integrated. The left side is ln
and the right side is -kt. Don't forget the constants:
ln
- ln(y_0) = -kt - 0 (y_0 is the initial population and the right hand side has a 0 because you integrate from 0 to t).
ln
= ln(y_0) - kt
y = e^(ln(y_0) - kt)
y = e^ln(y_0) * e^(-kt)
y = y_0 * e^(-kt)
That's the equation you had. What does it mean?
Your current population is equal to the initial population times some exponential function of time. As t increases, e^(-kt) will decrease and so the population will decrease. Since t starts at 0 and e^0 = 1, the population equals y_0 initially.
How do you relate this to your original equation?
Remember you had dy/dt = -ky. So what is dy/dt? You can differentiate your function!
dy/dt (y_0 * e^(-kt)) = y_0 * (-k) * e^(-kt)
Look at what you get: -k * (y_0 e^(-kt))
Remember that y = y_0 e^(-kt), so you really end up with -ky!!
Thus, dy/dt = -ky.
How do you use the function y = y_0 e^(-kt) ?
You need to know at least 2 values. Generally they tell you that at time t = 0 the population is such-and-such. If you plug in t = 0 you get y = y_0 so that population is the initial population. Then they might tell you that at time t = 3 (for example) the population is such-and-such. So you plug in that population, you know the time, and you know y_0. And you can get k. Finally, they ask you what the population is at time t = 5 (for example). Now that you know t, k, and y_0, you can solve for y.