Question

kathleen is a biologist studying the frogs at pinecrest lake. she is concerned about their decline, so she keeps track of their numbers. this table shows the frog population each year since kathleen started recording data.

years since initial recording (x)	population (y)
1	179
2	159
3	145
4	132
5	116

here is the exponential function that best fits the data. y = 198(0.9)^x
use the exponential function to complete the sentence below.
according to the function, the frog population at pinecrest lake will be 100 between

Explanation:

Step1: Set up the equation

We are given the exponential - decay function $y = 198(0.9)^x$, and we want to find $x$ when $y = 100$. So we set up the equation $100=198(0.9)^x$.

Step2: Isolate the exponential term

First, divide both sides of the equation by 198: $\frac{100}{198}=(0.9)^x$, which simplifies to $\frac{50}{99}=(0.9)^x$.

Step3: Take the natural logarithm of both sides

$\ln(\frac{50}{99})=\ln((0.9)^x)$. Using the property of logarithms $\ln(a^b)=b\ln(a)$, we get $\ln(\frac{50}{99}) = x\ln(0.9)$.

Step4: Solve for x

$x=\frac{\ln(\frac{50}{99})}{\ln(0.9)}$. We know that $\ln(\frac{50}{99})=\ln(50)-\ln(99)\approx3.912 - 4.595=- 0.683$ and $\ln(0.9)\approx - 0.105$. Then $x=\frac{-0.683}{-0.105}\approx6.5$.

Answer:

The frog population at Pinecrest Lake will be 100 between 6 and 7 years after the initial recording.