Answer :
The form of the exponential equation is
[tex]y=a(1+r)^x[/tex]a is the initial amount
r is the annual rate in decimal
Since the number of rabbits increases by 250% each year, then
[tex]r=\frac{250}{100}=2.5[/tex]Since the farm started with 12 rabbits, then
[tex]a=12[/tex]Substitute them in the form of the equation above
[tex]\begin{gathered} y=12(1+2.5)^x \\ y=12(3.5)^x \end{gathered}[/tex]We need to find the time for the rabbits to be 250
Then substitute y by 250 and solve the equation to find x
[tex]250=12(3.5)^x[/tex]Divide both sides by 12
[tex]\begin{gathered} \frac{250}{12}=\frac{12(3.5)^x}{12} \\ \frac{125}{6}=(3.5)^x \end{gathered}[/tex]Insert log to both sides
[tex]\log (\frac{125}{6})=\log (3.5)^x[/tex]Use the rule of the exponent with log
[tex]\log (a)^n=n\log (a)[/tex][tex]\log (\frac{125}{6})=x\log (3.5)[/tex]Divide both sides by log(3.5) to find x
[tex]\begin{gathered} \frac{\log(\frac{125}{6})}{\log(3.5)}=\frac{x\log (3.5)}{\log (3.5)} \\ 2.423885719=x \end{gathered}[/tex]It will take about 2.42 years to be 250 rabbits