Subsections

• Introduction
• Background
• Exercises

Exponential, Logarithmic, and Logistic Functions

Introduction

The purpose of this lab is to use Maple to study exponential, logarithmic, and logistic functions. These are used to model many types of growth, as well as in many scales, such as the Richter and decibel scales.

Background

Logarithmic Scale: The decibel scale is a logarithmic scale used to measure sound. Measured at some point A, the measurement of the sound in decibels is

$$D(A)=10\log_{10} \left(\frac{I(A)}{I_0}\right)$$

where I(A) is the sound intensity at the point A, and I₀ is a reference intensity chosen so that the threshold of human hearing is approximately 0 decibels.

Exponential growth: The simplest model for growth is exponential, where it is assumed that y'(t) is proportional to y. That is,

$$\frac{dy}{dt} = ky\quad\hbox{where $k$\space is a positive constant.}$$

Separating the variables and integrating (see section 4.4 of the text), we have

$$\int\frac{dy}{y} = \int k\,dt$$

so that

$$\ln y = kt + C$$

(If y is a positive quantity, we may drop the absolute value signs around y.) Solving for y

y = e^{kt + C}

which we may write in the form y = Ae^kt, where A is an arbitrary positive constant. (Why?)

Exponential Decay: In a sample of a radioactive material, the rate at which atoms decay is proportional to the amount of material present. That is,

$$\frac{dA}{dt} = -kA$$

where k>0 is a constant. This is the same equation as in exponential growth, except that -k replaces k. The solution is

A(t) = A₀ e^-kt

where A₀ is a positive constant. Physically, A₀ is the amount of material present at t=0.

Radioactivity is often expressed in terms of an element's half-life. For example, the half-life of carbon-14 is 5730 years. This statement means that for any given sample of ${}^{14}\hbox{C}$, after 5730 years, half of it will have undergone decay. So, if the half-life is of an element Z is c years, it must be that $e^{-kc}=\frac{1}{2}$, so that $kc=\ln 2$ and $k=\frac{\ln 2}{c}$.

Logistic growth: One of the consequences of exponential growth is that $y\rightarrow\infty$as $t\rightarrow\infty$. However, in some situations there is a limit B to how large y can get. For example, the population of bacteria in a laboratory culture, where the food supply is limited. In such situations, the rate of growth slows as the population reaches the carrying capacity. One useful model is the logistic growth model. It assumes that the rate of growth is proportional to the product of the population and the difference between the population and its upper limit. Thus we model the growth with the differential equation

$$\frac{dy}{dt} = ky(B-y).$$

In the exercises you will use Maple to solve this equation and work with an example.

Exercises

1.

Suppose element Z has a half-life of 2 years.

(a)

Choose an initial amount A₀ for the element. Find the amount as a function of time, and plot this function.

(b)

How long is it until the amount is $\frac{1}{3}$ of what it was initially?

(c)

How long is it until the amount is $\frac{1}{4}$ of what it was initially? Is there a quick way to answer this question?

2.

In 1935 Charles F. Richter of Cal Tech developed a scale for measuring the magnitude of earthquakes. The Richter Scale formula is given by

$$M = \log_{10} \left( \frac{x}{c} \right)$$

where M is the magnitude of the earthquake, x is the amplitude of the largest seismic wave as measured on a standard seismograph 100 kilometers from the epicenter and c is the amplitude of a reference earthquake of amplitude 1 micron ( 1 micron is 0.001 mm) on a standard seismograph at the same distance from the epicenter.

(a)

When the amplitude of an eqrthquake is tripled, by how much does the magnitude increase?

(b)

In 1989, the San Francisco Bay area suffered severe damage from an earthquake of magnitude 7.1. However, the damage was not nearly as extensive as that caused by the great quake of 1906, which has been estimated to have had the magnitude 8.3. What is the ratio of the amplitude of the 1906 quake to the 1989 quake?

(c)

The largest earthquake magnitude ever measured was 8.9 for an earthquake in Japan in 1933. Determine the ratio of the amplitude of this earthquake to that of the 1906 San Francisco earthquake.

3.

Consider the logistic growth differential equation

$$\frac{dy}{dt} = ky(B-y).$$

Separating the variables yields the equation

$$\int \frac{dy}{ky(B-y)} = \int dt.$$

(a)

Use Maple to evaluate these integrals. Then use Maple's results to solve the differential equation for y as a function of x. (Remember that maple does not include a constant of integration in its answer, so you will have to include it yourself. Note, also, that maple does not include absolute-value signs when integrating $\int \frac{dy}{y}$, so you may have to modify the arguments of logarithms, to make sure that you are taking the log of a positive number. Be sure to explain clearly what you are doing.)

(b)

Explain how your answers to (a) may be written in the form

$$y = \frac{B}{1+Ae^{-Bkt}},$$

where A is some positive constant.

(c)

Suppose at time t=0 the population of bacteria in a culture is 1000, and the limited environment can support no more than 100,000 bacteria. After 1 hour the population is 5,000. Find the values of A and k. Then plot y as a function of t. What is the population when the rate of growth is most rapid?