Solution:
The population of cockroaches grows exponentially. 2 months ago there were 3 cockroaches, now there are 18.
As this is exponential growth, it can be modeled with the following formula:
[tex]y(t)=a.(e^k)^t[/tex]
Applying the laws of exponents, this is equivalent to:
[tex]y(t)=a.e^k^t[/tex]
where y(t) is the number of cockroaches at time t, a is the initial population. In this case, we want to find k.
Now, we know a=3, t=2, and right now y(2)=18:
[tex]18=3.e^{2k}[/tex]
this is equivalent to:
[tex]6=e^{2k}[/tex]
Take the natural logarithm of both sides:
[tex]\ln (6)=\ln (e^{2k})[/tex]
this is equivalent to:
[tex]\ln (6)=2k[/tex]
solving for k, we get:
[tex]k\text{ =}\frac{\ln (6)}{2}[/tex]
Therefore, we have created a real-life problem that uses the laws of exponents.
