Question

select the correct answer.
a geneticist needs to grow a stock of fruit flies for her experiments. she currently has a stock of 200 fruit flies and predicts the stock will grow by 38% each day. which function could she use to calculate ( f(n) ), the number of days required for her stock to grow to ( n ) fruit flies?
a. ( f(n)=log _{0.38}left(\frac{200}{n}
ight) )
b. ( f(n)=log _{1.38}left(\frac{n}{200}
ight) )
c. ( f(n)=log _{1.38}left(\frac{200}{n}
ight) )
d. ( f(n)=log _{0.38}left(\frac{n}{200}
ight) )

Explanation:

Step1: Write the exponential growth formula

The general formula for exponential growth is \(n = a(1 + r)^t\), where \(a = 200\) (initial amount), \(r=0.38\) (growth rate), \(t\) is the number of days, and \(n\) is the final amount. So, \(n = 200(1 + 0.38)^t\).

Step2: Solve for \(t\)

First, divide both sides by \(200\): \(\frac{n}{200}=(1.38)^t\).
Then, use the definition of a logarithm. If \(y = b^x\), then \(x=\log_b y\). Here, \(b = 1.38\), \(x=t\), and \(y=\frac{n}{200}\). So, \(t=\log_{1.38}(\frac{n}{200})\).

Answer:

B. \(f(n)=\log_{1.38}(\frac{n}{200})\)