To determine which logarithmic function models the time, [tex]\( f(n) \)[/tex], in years it will take for the number of amphibian species to decrease to a value of [tex]\( n \)[/tex], given that the current number of species is 74 and the rate of decrease is 2% per year, follow the steps below:
1. Understand the problem parameters:
- Current number of species ([tex]\( S_0 \)[/tex]) = 74
- Annual rate of decrease = 2%
2. Express the rate of decrease:
Since the species decrease by 2% each year, 98% of the species remain each year. Thus, the annual decay factor is:
[tex]\[
\text{Decay factor} = 1 - 0.02 = 0.98
\][/tex]
3. Formulate the exponential decay model:
To understand how the number of species changes over time, we'll use the exponential decay formula:
[tex]\[
S(t) = S_0 \cdot (0.98)^t
\][/tex]
Where [tex]\( S(t) \)[/tex] is the number of species after [tex]\( t \)[/tex] years.
4. Set up the equation to solve for [tex]\( t \)[/tex]:
We want to find the time [tex]\( t \)[/tex] when the number of species is [tex]\( n \)[/tex]. So, set [tex]\( S(t) = n \)[/tex]:
[tex]\[
n = 74 \cdot (0.98)^t
\][/tex]
5. Isolate the exponential term:
Divide both sides by 74 to isolate the exponential term:
[tex]\[
\frac{n}{74} = (0.98)^t
\][/tex]
6. Use the logarithm to solve for [tex]\( t \)[/tex]:
To solve for [tex]\( t \)[/tex], take the logarithm with base 0.98 on both sides:
[tex]\[
\log_{0.98} \left(\frac{n}{74}\right) = t
\][/tex]
Rearranging this gives us the function:
[tex]\[
t = \log_{0.98} \left(\frac{n}{74}\right)
\][/tex]
7. Identify the correct option:
Comparing this logarithmic function with the given options, we find that the function:
[tex]\[
f(n) = \log_{0.98} \left(\frac{n}{74}\right)
\][/tex]
matches option C.
Therefore, the correct answer is:
[tex]\[ \boxed{f(n)=\log _{0.98}\left(\frac{n}{74}\right)} \][/tex]
