We have a function with an exponetial form:
[tex]\begin{gathered} f(t)=a\cdot b^t \\ In\text{ this case:} \\ f(t)=54\cdot2.718^{0.12t} \\ \text{Where t is the time in month} \end{gathered}[/tex]
The initial value is when t=0 (means 0 month), so:
[tex]\begin{gathered} f(t=0)=a\cdot b^0=a,b^0=1\text{ for any b} \\ f(t=0)=54\cdot2.718^{0.12\cdot0}=54 \end{gathered}[/tex]
When the researcher started the number of birds was 54. This discard the second and the last option.
And, we can find the concept of b taking the ratio of f(t) for two different values of t:
[tex]\begin{gathered} We\text{ say:} \\ t_2>t_1\ge0 \\ f(t_2)=a\cdot b^{t_2} \\ f(t_1)=a\cdot b^{t_1} \\ \frac{f(t_2)_{}}{f(t_1)}=\frac{a\cdot b^{t_2}}{a\cdot b^{t_1}}=b^{(t_2-t_1)},t_2-t_1>0 \end{gathered}[/tex]
So, the base b is related with the growth of the population, if b is greater than 1 the population increase with time, if b is less than 1 the population decrease with time.
So, the correct answer is the first option.
