Answer:
Proof below
Step-by-step explanation:
Exponential Grow Model
The equation to model some time dependant event as an exponential is
[tex]A=A_oe^{kt}[/tex]
Where Ao is the initial value, k is a constant and t is the time. With the value of Ao and k, we can compute the value of A for any time
We are required to find the time when the population being modeled doubles from Ao to 2 Ao. We need to solve the equation
[tex]2A_o=A_oe^{kt}[/tex]
Simplifying by Ao
[tex]2=e^{kt}[/tex]
Taking logarithms in both sides
[tex]ln2=lne^{kt}[/tex]
By properties of logarithms and since lne=1
[tex]ln2=kt\cdot lne=kt[/tex]
Solving for t
[tex]\displaystyle t=\frac{ln2}{k}[/tex]
Hence proven
