How To Study with DWFK Precalculus

Chapter 3: Exponential, Logistic, and Lograithmic Functions

This chapter begins the study of transcendental functions, once a significant threshold for most algebra students to cross. In taking this course, you will have already encountered four such functions among the Twelve Basic Functions of Chapter 1 (and, in fact, will have analyzed most of their important properties), so the threshold will already have been crossed.

The emphasis in this chapter will be on the particular algebraic properties of these functions and on the kinds of real-world behavior they model.

Section 3.1 Section 3.4

Section 3.2 Section 3.5

Section 3.3 Section 3.6

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Section 3.1 Exponential and Logistic Functions

Objectives
You will be able to evaluate exponential expressions and be able to identify and graph exponential and logistic functions.

Key Ideas

Algebraic function Exponential growth and decay factors

Base (of an exponential function) Limit to (logistic) growth

Basic logistic function Logistic function

Exponential function Logistic growth and decay

Exponential growth and decay The natural base e

Study Tips
This section is intended as a genuine introduction to these important classes of functions (as opposed to a re-hashing of the algebra of exponents). In keeping with the theme of the course, the examples look at the functions algebraically, graphically, and numerically.

There is very little redundancy in the examples and explorations, each of which reveals something important about the functions.

Technology Tips
Use your calculators early and often in this section so that they become familiar with the characteristic shapes of exponential and logistic functions. (Exploration 1 is particularly useful in this regard.) The logistic functions can look a little intimidating algebraically, so students will need the graphical "hook" to get acquainted with them.

Note when setting the window for a logistic growth function that the horizontal window should contain the y-axis. The vertical window should contain the x-axis and the horizontal asymptote y = c (where c is the limit to growth).
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Section 3.2 Exponential and Logistic Modeling

Objectives
You will be able to model real-world problems involving exponential and logistic growth and decay. You will be able to convert among algebraic, graphical, and numerical models for exponential and logistic growth.

Key Ideas

Constant percentage rate Maximal sustainable population

Half-life Radioactive decay

Study Tips
The modeling applications of exponential and logistic functions are many and varied. The precalculus courses of years past concentrated on the algebraic properties of exponents (understandable, since facility in handling these was critical for solving equations), deferring the applications until calculus. Today, however, modern cries for more relevant courses have turned the attention to modeling, while technology has thankfully freed up some equation-solving time to enable your teachers to respond to the cries.

Technology Tips
There are many calculators that do logistic regression, but it is unfortunate that they do not all use the same logistic function. The logistic functions in this textbook solve the logistic differential equation (encountered in calculus) , an equation that leads to a curve with a horizontal asymptote at y = 0. Other calculators use a model with a parameter for vertical translation, allowing for non-zero horizontal asymptotes both above and below the curve. This latter strategy allows for a better "fit" for some data, but it does not yield a curve that solves the logistic differential equation.

In calculus (and therefore in precalculus) courses, we believe that it is more important to solve the differential equation than to follow the shape of an arbitrary set of data points. Happily, the best-selling calculator models agree with us. Be advised, though, that if your students have trouble getting our function when attempting logistic regression, the fault might lie with their calculators and not with themselves.

Remember when doing exponential and logistic regressions that the data in the y list must be positive.

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Section 3.3 Logarithmic Functions and Their Graphs

Objectives
You will be able to convert equations between logarithmic form and exponential form, evaluate common and natural logarithms, and graph general logarithmic functions.

Key Ideas

Common logarithm Logarithmic function

Decibel Natural logarithm

Study Tips
Understanding the inverse relationship between logarithmic functions and exponential functions is the key to understanding their algebraic and graphical properties. This section concentrates on that relationship, making it a good review of the general concept of inverse functions.

Technology Tips
This section is a good example of the authors' intent to incorporate the use of the graphing calculator into the text in a natural way while keeping the focus on the mathematics. There is very little in this section that does not invoke technology in some way, and yet the object is to teach you about logarithms, not about calculators. By this point, one-third of the way through the book, you and your colleagues should be using technology at appropriate times without really thinking much about it.
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Section 3.4 Properties of Logarithmic Functions

Objectives
You will be able to apply the properties of logarithms to evaluate expressions, change bases, and re-express data.

Key Ideas

Change-of-base formula for logarithms Properties of logarithms

Kepler's Third Law Re-expression of data

Study Tips
You might well have seen the algebraic properties of logarithms in a previous algebra course, but we do not assume that. This section, an introduction for some and a review for others, provides a good preparation for the equation-solving to come in Section 3.5.

Technology Tips
Although, lamentably, some algebraic skills have atrophied in your generation since the arrival of equation-solving technology, the change-of-base formula for logarithms is an interesting example that runs counter to that trend. You still need to know the change-of-base formula to find (for example) log₃16, even with calculators, but with calculators you can actually arrive at a quick answer. This immediate gratification has made you into a generation of algebra students who can change logarithm bases as easily as you can solve quadratic equations, often with greater reliability.
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Section 3.5 Equation Solving and Modeling

Objectives
You will be able to apply the properties of logarithms to solve exponential and logarithmic functions algebraically. You will be able to solve applied problems using these equations.

Key Ideas

Newton's Law of Cooling pH

Order of magnitude Richter scale

Study Tips
This section incorporates some of the better-known applications of exponential and logarithmic equations in order to give you contextual practice in solving them. You will get valuable experience using the functions while gaining insights into the many ways they can be used to model real-world behavior.

Newton's Law of Cooling is often seen in calculus courses as a solution to a problem in differential equations. Here, we give them the formula, making it appropriate for precalculus students.

Technology Tips
We hope that you will not have to spend too much time in this section learning the material from physics and chemistry. This is always a risk with applications. The emphasis here is on modeling, requiring some explanation of the various models, but in the end you will simply have to accept such formulas as pH and the Richter scale without much justification.

Remember when doing logarithmic regression that the data in the x list must be positive.

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Section 3.6 Mathematics of Finance

Objectives
You will be able to use exponential functions and equations to solve business and finance problems related to compound interest and annuities.

Key Ideas

Annual percentage rate (APR) Compounded continuously

Annual percentage yield (APY) Future value of an annuity

Annuity Ordinary annuity

Compound interest Present value of an annuity

Study Tips
It is probably not difficult to motivate you to solve problems involving money, as you can easily appreciate their relevance. This section is a little less important for calculus and a little more important for life.

The annuity formulas are given here because of their importance in the world of finance, despite the fact that we cannot justify the formulas until we cover geometric series. While it is not recommended that you memorize these formulas, you should be able to use them to solve problems in the section exercises.

Technology Tips
There are modern calculators that have user-friendly financial packages built in (including compound interest and annuities), but using them here deprives you of the practice that you might otherwise get using exponential and logarithmic functions.