Mathematics 101 Transitional Mathematics

Study Guide :: Unit 5

Exponential and Logarithmic Functions

Exponential expressions are distinguished by their notation, which was originally intended to be a simplified way of representing numbers having the form: a single natural number (such as 1, 2, 3, 4, 5, . . .) multiplied together repeatedly. Perhaps because counting how many 2’s are multiplied together in the following product 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 · 2 is a rather tedious, time-consuming chore, a new notation was developed to avoid this dull repetitive presentation which, while correct, was not easy to read. Enter the special number, called an exponential, looking like ${\displaystyle 2^{30}}$, instead of the previous repetitive multiplication sequence. This is read as the base number ${\displaystyle a=2}$ multiplied together 30 times.

Exponentials are also expressible as functions if we keep the base ${\displaystyle a}$ constant but vary the exponents. Therefore, ${\displaystyle a^x}$ became the notation for an exponential function where the base ${\displaystyle a}$ is a constant and the exponent ${\displaystyle x}$ is the independent variable. This is certainly the first function in this course in which the variable is not on the fraction line. Note the difference in the position of the variable ${\displaystyle x}$ between the exponential functions ${\displaystyle a^x}$, the root functions ${\displaystyle a\sqrt[n]{x}=a{x}^{\frac{1}{n}}}$, and power functions, ${\displaystyle a{x}^n}$, the latter being a special case of polynomials called the monomials.

Now let’s go back to the origins of the notion of a ‘logarithm’—literally invented by a Scottish Baron, Lord John Napier (1550–1617), at a time which pre-dated even the idea of a mathematical function. When Napier first began to calculate his ‘logarithms,’ he had called them ‘artificial numbers.’ Why artificial? Simply because the decimal notation we commonly use today was just being introduced to the European scientific community by one of Napier’s contemporaries—a Flemish mathematician and military engineer called Simon Stevin (1548–1620). Moreover, even the decimal notation itself, in which the point separates the integers from the decimal fractions, seems to have been the invention of another of Napier’s contemporaries, a German/Polish mathematician, astronomer, and theologian called Bartholomaeus Pitiscus (1561–1613). Pitiscus had used this particular decimal notation, taken to five or six decimal places, in the calculation of his trigonometric tables (circa 1600). John Napier subsequently adopted Pitiscus’s notation in the writing of in his logarithmic papers published in 1614 and 1619 (posthumously). As is often the case with new discoveries or inventions, acceptance of these ‘artificial’ numbers was lukewarm and there was considerable resistance to recognize them as valid numbers by the scientific community of the time, with the notable exception of Johannes Kepler (1571–1630), another German mathematician and astronomer.

So what exactly are these numbers which seemed to have been created so artificially by Napier? It is important to note that the notion of a real number, at least those expressible as non-repeating decimals, was not formalized into mathematical existence until the latter part of the nineteenth century. Richard Dedekind (1831–1916) laid the foundations of the real number system. If we return to our modern way of looking at exponential functions, Napier’s logarithm (to a given constant base ${\displaystyle a}$) was simply the exponential process in reverse. In functional notation, it looks like this

${\displaystyle {\exp }_a\text{:}\hspace{0.5em}x\mapsto a^x=N}$ is the exponential process
${\displaystyle {\log }_a\text{:}\hspace{0.5em}{a}^x=N\mapsto x}$ is the logarithmic process.

From these beginnings, it is now known that various unit measurements in scientific or science-related disciplines are expressed as logarithms of other quantities. For example, the pH factor in chemistry, the apparent magnitude of the stars from our earthly perspective, the decibel in telecommunications and engineering, the Richter scale magnitude of earthquakes, and semitones in music are all defined using logarithms to various bases. Once again the development of mathematical concepts and ideas, initially considered useless, has led to scientific expressions which mirror processes in our real world. Because of the pre-eminence of the logarithm in some circles, the exponential process is also known as the anti-logarithm, more commonly spelt antilogarithm.

Learning Objectives

After you have completed Unit 5, you should be able to:

1. define and give examples of exponentials and logarithms;
2. define exponentials and logarithms (with a given fixed base) as functions of a single variable and recognize their interrelationship as functions;
3. determine the domains, ranges, fundamental properties, values and graphs of exponential and logarithm functions;
4. distinguish between an exponential function and a power function (a polynomial with a single term);
5. list the fundamental properties of exponentials and logarithms to any base, calculate their values and perform operations on them as algebraic expressions;
6. distinguish between the natural logarithm and other logarithms;
7. convert between logarithms of different bases;
8. solve equations involving exponentials and logarithms; and
9. apply exponentials and logarithms to real-world situations.

Readings and Practice

Chapter 5 (pp. 161–210) of the textbook

Optional Section

• Focus on Modeling: Fitting Exponential and Power Curves to Data (pp. 217–226)

Online logarithm/anti-logarithm (or exponential) calculator

Online graphing tool: Graph

Exponential Functions

Textbook Readings

pp. 162–167

Practice

(pp. 167–169):  1, 3, 4, 9, 11, 15, 19, 21, 25, 33, 39, 47, 49, 53, 55

Answers

pp. 510–511

Terms to Understand

formal definition of an exponential function; power function; compound interest formula

Figures/Tables

Exponential Functions with a specified base (definition) (p. 162)
Graphs of Exponential Functions (p. 164)
Compound Interest (compounded in discrete intervals of time) (p. 166)

Online Tutorials—the Overview

Pertinent to “Exponential Functions”

• Exponentials: an introduction
• Exponentials with natural number exponents
• Exponentials with positive and negative whole number exponents
• Numerical Roots as Exponentials
• Exponentials with fractional exponents

Online Maple TA Practice

See links under the top menu ‘Assessment’ tab of the particular topic page.

The Natural Exponential Function

Textbook Readings

pp. 170–172

Practice

(pp. 172–175):  1, 2, 5, 7, 9, 15, 21, 27, 31

Answers

pp. 511–512

Terms to Understand

formal definitions of the natural logarithm; continuous compound interest formula

Figures/Tables

The Natural Exponential Function (p. 170)
Compound Interest (continuously compounded) (p. 172)

Logarithmic Functions

Textbook Readings

pp. 175–181

Practice

(pp. 182–184):  1–4, 5, 7, 11, 13, 15, 17, 23, 25, 31, 41, 47, 51, 53, 55, 59, 61, 65, 71, 75, 85, 89

Answers

pp. 512–513

Terms to Understand

formal definitions of logarithm functions to various fixed bases, the common logarithm, the natural logarithm, and their properties

Figures/Tables

Definition of the Logarithmic Function (p. 175)
Properties of Logarithms (p. 176)
The Common Logarithm (p. 179)
The Natural Logarithm (p. 180)
Properties of the Natural Logarithm (p. 181)

Online Tutorials—the Overview

Pertinent to “Logarithmic Functions”

• Logarithms: an introduction
• Logarithms: definitions and evaluation, including antilogarithms

Online Maple TA Practice

See links under the top menu ‘Assessment’ tab of the particular topic page.

Laws of Logarithms

Textbook Readings

pp. 185–189

Practice

(pp. 189–191):  1–6, 7, 11, 13, 17, 19, 25, 29, 35, 37, 39, 45, 49, 53, 57, 65, 71

Answers

p. 513

Terms to Understand

product, quotient, and power rules of logarithms; change of base formula for logarithms

Figures/Tables

Laws of Logarithms (p. 185)
Change of Base Formula (p. 188)

Online Tutorials—the Overview

Pertinent to “Laws of Logarithms”

• Logarithms: properties

Online Maple TA Practice

See links under the top menu ‘Assessment’ tab of the particular topic page.

Exponential and Logarithmic Equations

Textbook Readings

pp. 191–198

Practice

(pp. 198–199):  1, 2, 3, 5, 7, 13, 17, 21, 25, 29, 33, 37, 43, 45, 51, 55, 59, 67, 69, 71, 75, 79, 85

Answers

p. 514

Terms to Understand

formal definitions of an exponential equation and a logarithmic equation

Figures/Tables

Guidelines for Solving Exponential Equations (p. 191)
Guidelines for Solving Logarithmic Equations (p. 194)

Online Tutorials—the Overview

Pertinent to “Exponential and Logarithmic Equations”

• None

Online Maple TA Practice

See links under the top menu ‘Assessment’ tab of either the ‘Exponentials’ or ‘Logarithms’ topic pages.

Modeling with Exponential and Logarithmic Functions

Textbook Readings

pp. 200–210

Practice

(pp. 210–213):  1, 3, 7, 15, 17, 27, 35

Answers

p. 514

Terms to Understand

exponential growth; radioactive decay; logarithmic scales

Figures/Tables

Exponential Growth (Doubling Time) (p. 201)
Exponential Growth (Average Growth Rate) (p. 202)
Radioactive Decay Model (p. 205)

Online Tutorials—the Overview

Pertinent to “Modeling with Exponential and Logarithmic Functions”

• Logarithms: applications in finance

Online Maple TA Practice

See links under the top menu ‘Assessment’ tab of the particular topic page.

Unit 5 Review

(pp. 213–215):  odd numbers

Answers

pp. 514–515

Unit 5 Test

(p. 216):  1–10

Answers

pp. 501–502

Figures/Tables

Formulas to Remember (pp. iv and v at the beginning of the text)

Midterm Exam Preparation

Covering Units 1 through 5

Skills which are assumed:

Unit 1 and Unit 2

It is assumed that your basic algebra skills are good, if not excellent, and that you can read a word problem and analyse it for what is given and what you must find in mathematical terms.

Skills to be tested:

Unit 3

Analysis of a general function with regards to

1. its properties: domain, its range, its roots/zeroes, its asymptotes (both vertical and horizontal), its graph;
2. how to operate on functions (add, subtract, multiply, divide, compose, and invert) and how to analyse the resulting functions from those operations;
3. how to find the average rate of change of a function and what that means in terms of its graph and its measurements (if any);
4. how to transform functions (vertical/horizontal translations, contraction/expansion scaling, reflections over the major axes); and
5. how to use functions in real-life situations and how to solve problems associated with them.

Unit 4

Analysis of a polynomial or rational function with regards to

1. its properties: domain, its range, its roots/zeroes, its asymptotes (both vertical and horizontal), its graph;
2. how to operate on them (add, subtract, multiply, divide, factor, compose, and invert) and how to analyse the resulting functions from those operations;
3. the factor theorem, the remainder theorem and the fundamental theorem of algebra and their interconnections with polynomials, their factors and their divisors;
4. how to find the average rate of change of a polynomial/ rational function and what that means in terms of its graph and its measurements (if any);
5. how to transform polynomials/ rational functions (vertical/horizontal translations, contraction/expansion scaling, reflections over the major axes);
6. how to solve polynomial/ rational function equations and inequalities; and
7. how to use polynomials/ rational functions in real-life situations and how to solve problems associated with them.

Unit 5

Analysis of an exponential or logarithm function with regards to

1. its properties: domain, its range, its roots/zeroes, its asymptotes (both vertical and horizontal), its graph;
2. how to operate on them (add, subtract, multiply, divide, factor, compose, and invert) and how to analyse the resulting functions from those operations;
3. how to find the average rate of change of an exponential/logarithm function and what that means in terms of its graph and its measurements (if any);
4. how to transform exponential/logarithm functions (vertical/horizontal translations, contraction/expansion scaling, reflections over the major axes);
5. how to solve exponential/logarithm function equations and inequalities; and
6. how to use exponential/logarithm functions in real-life situations and how to solve problems associated with them.

Extra Practice

Take a random sample of questions from each of the unit sections below and try to solve them without looking at the text materials. That will give you some idea as to how well you have remembered and understood the concepts and procedures.

Midterm Pretest

Textbook

(pp. 227–228):  1–12   (In #6a, you do not need to find the complex roots.)

Answers

pp. 515–516