calculus - why do I have to double an exponential growth and half an exponential decay function to find out how fast/slow the process is?

Let’s take the exponential growth function to be $y= A_0e^{kt}$ and exponential decay function to be $y= A_0e^{-kt}$ .

I’m just told that to find out How fast the process is, you have to double the exponential growth function and Half-time the exponential decay function. Why is this so? I never seem to understand it. Is there a way to prove it? Or is my tutor right by just telling me the fact and nothing else?

Answer

Let's take the exponential growth function $f(t)=A_0e^{kt}$. It is simple to find $A_0$ - this is just the value of $f$ at time $0$. But how can we find the value of $k$ ?

Suppose $f(t)$ has value $y_0$ at time $t_0$ and value $2y_0$ at time $t_1$. Then

$2y_0=2A_0e^{kt_0}=A_0e^{kt_1}$

$\Rightarrow 2e^{kt_0}=e^{kt_1}$

$\Rightarrow \ln(2) + kt_0=kt_1$

$\Rightarrow k=\frac{\ln(2)}{t_1-t_0}$

The time interval $t_1-t_0$ during which $f(t)$ doubles is called the "doubling time". Notice that it does not depend on when it is measured. Once you know the doubling time you can find the growth constant $k$.

You can do the same calculation with exponential decay except you measure the time taken for $f(t)$ to halve, not double, which is called the "half life".