Answer:
The function [tex]f(n)=log_{0.96}(\frac{n}{186})[/tex]
Explanation:
Wherever we are studying logarithmic decrease or increase, we can construct our logarithmic function following the next recipe :
[tex]f(n)=log_{x}(\frac{n}{y})[/tex] (I)
Where [tex]''n''[/tex] is the value that we want to obtain (following a logarithmic decrease or increase).
Where [tex]''f(n)''[/tex] is the time needed to obtain that variation.
Where [tex]''x''[/tex] is the parameter related to the percentage decrease or increase
And where [tex]''y''[/tex] is known as the original population.
Now let's use the question to learn how to use the expression (I) :
In order to find the parameter [tex]''x''[/tex] we need to know how porcentual the population changes. We know that the number of insect species decrease by 4 % per month.
Therefore, for the initial month we will have the 100 % of the number of insect species. For the first month we will have a decrease of 4 % which can be written as the 96 % of the original population.
We write 100 % - 4 % = 96 % ⇔ [tex]1-0.04=0.96[/tex]
That's how we obtain the parameter [tex]x[/tex]. In this case we subtract 0.04 to the original population 1 because we have a decrease. Otherwise we would have added if we had had an increase. The number 0.96 represents that month by month we obtain the 96 % from the previous month. The value from the parameter is [tex]x=0.96[/tex]
Now the original population of insect species is [tex]186[/tex]. Therefore the value for [tex]y[/tex] is [tex]y=186[/tex]. Using the values obtained in the expression (i) we have :
[tex]f(n)=log_{0.96}(\frac{n}{186})[/tex]
For example,
We have [tex]186[/tex] number of insect species at the time origin. A decrease of 4 % can be calculated as
[tex](186)(0.96)=178.56[/tex] which is the value expected for the first month. Now if we use this value in the expression (I) :
[tex]n=178.56[/tex] ⇒ [tex]f(178.56)=log_{0.96}(\frac{178.56}{186})=log_{0.96}(0.96)=1[/tex] ⇒
[tex]f(178.56)=1[/tex]
This means that to obtain 178.56 number of insect species we need to wait one month to achieve this value. Which fits with the problem data.
