Let's solve this step-by-step.
1. Understanding the Problem:
- The number of amphibian species in a rain forest decreases by 2% per year.
- The current number of species is 74.
- We need to model the time, [tex]\( f(n) \)[/tex], it takes for the number of species to decrease to a value of [tex]\( m \)[/tex] using a logarithmic function.
2. Setting up the Exponential Decay Model:
- The initial number of species is [tex]\( 74 \)[/tex] (denoted as [tex]\( n_0 \)[/tex]).
- The rate of decrease per year is [tex]\( 2\% \)[/tex] or [tex]\( 0.02 \)[/tex].
- The decay factor per year is [tex]\( 1 - 0.02 = 0.98 \)[/tex].
The exponential decay model can thus be written as:
[tex]\[
n(t) = 74 \times (0.98)^t
\][/tex]
where [tex]\( n(t) \)[/tex] is the number of species remaining after [tex]\( t \)[/tex] years.
3. Finding the Time [tex]\( t \)[/tex]:
- We need to find the time [tex]\( t \)[/tex] when the number of species [tex]\( n(t) \)[/tex] decreases to [tex]\( m \)[/tex].
Rearranging the decay model equation:
[tex]\[
m = 74 \times (0.98)^t
\][/tex]
To isolate [tex]\( t \)[/tex], we'll take the logarithm base [tex]\( 0.98 \)[/tex] on both sides:
[tex]\[
\log_{0.98}(m) = \log_{0.98}(74 \times (0.98)^t)
\][/tex]
4. Using Logarithmic Properties:
- Applying the property of logarithms: [tex]\( \log_b(a \times c) = \log_b(a) + \log_b(c) \)[/tex]:
[tex]\[
\log_{0.98}(m) = \log_{0.98}(74) + \log_{0.98}((0.98)^t)
\][/tex]
[tex]\[
\log_{0.98}(m) = \log_{0.98}(74) + t
\][/tex]
- Rearranging for [tex]\( t \)[/tex]:
[tex]\[
t = \log_{0.98}(m) - \log_{0.98}(74)
\][/tex]
[tex]\[
t = \log_{0.98}\left(\frac{m}{74}\right)
\][/tex]
5. Conclusion:
- The logarithmic function that models the time, [tex]\( f(n) \)[/tex], in years, to decrease the number of species to [tex]\( m \)[/tex], is:
[tex]\[
f(n) = \log_{0.98}\left(\frac{n}{74}\right)
\][/tex]
Thus, the correct answer is [tex]\( \boxed{\text{C}} \)[/tex].
