Section 6.7 - Applications of Exponential and Logarithmic Functions

Section Objectives

1. Solve application problems involving exponential and logarithmic equations.
2. Determine an exponential or logarithmic model for a problem situation.

Examples

• In a research experiment, a population of fruit flies is growing exponentially. After 2 days there are 100 flies, and after 4 days there are 300 flies. Find a model of the form $P(t)=Ae^{kt}$ that describes the population at time $t$. How many flies will there be after 5 days?

• When using carbon-14 dating, scientists use the formula $\displaystyle R=\frac{1}{10^{12}}e^{-t/8223}$, where $R$ is the ratio of carbon-14 to carbon-12 $t$ years after death. Estimate the age of an organic object for which the ratio of carbon-14 to carbon-12 is $1/10^{13}$.

• Plutonium-239 has a half-life of about 24,100 years. Use a model of the form $P(t)=Ae^{-kt}$ to determine the initial amount of Pu-239 if there were 0.4 grams remaining after 1000 years.

• One-hundred animals were released into a preserve where their population grows according to the model $\displaystyle P(t)=\frac{1000}{1+9e^{-0.1656t}}$, where $t$ is measured in months. After how long will the population reach 750 animals?