13.16: How Managers Can Encourage Ethical Behavior
- Explain the purpose of a code of ethics (why US laws require one for companies listed on the largest stock exchanges).
- Explain the benefits of ethics training.
- Describe the methods of selecting and hiring ethical employees.
- Explain whistleblower protection.
- Explain senior management’s role in fostering ethical decisions and behavior.
Codes of Ethics
A code of ethics , also known as a code of conduct or statement of values, is a policy statement of a company’s values, responsibilities, and conduct expectations. The purpose of a code of ethics is to guide employees in handling ethical dilemmas. It is essentially a moral compass.
The ethical lapses of global companies, such as Enron, Arthur Andersen, and WorldCom, over the span of a few years caused much devastation. Enron and WorldCom executives used deceptive accounting and sold inflated stock while recommending it to employees and external investors. Arthur Andersen’s accountants, who audited Enron and WorldCom, turned a blind eye to the fraud. As a result, in 2002, Congress passed the Sarbanes-Oxley Act, which requires that companies listed or applying to be listed on a public stock exchange establish and enforce a code of ethics. It also requires that any changes to an established code of ethics be disclosed to the public. A code of ethics that meets the requirements of the act comprises the standards necessary to promote “honest and ethical conduct; full, fair, accurate, timely and understandable disclosure in periodic reports” and “compliance with applicable governmental rules and regulations.” [1]
Codes of ethics vary in content, length, and complexity. Johnson & Johnson has a relatively simple code of conduct, with just four paragraphs that can be summarized as: Our customers are our first priority, then our employees, then our communities. Stockholders come last. But if we take care of the others, then stockholders will do well. [2] Without being too detailed or lengthy, this credo expresses the company’s values and gives employees the guidance they need to make ethical decisions.
Ethics Training
Having codes and policies in place that address ethics is not enough. Employees need to be taught how to respond in situations involving ethics. Therefore, many managers enroll their employees in an ethics training program. Ethics programs often involve activities that encourage ethical behavior and reinforce a company’s ethics code/policies.
One ethics activity involves having employees match various scenarios on one set of cards with the possible responses on another set of cards. Every scenario will have appropriate and inappropriate answers. After the participants match scenarios with responses, the group discusses the decision. Discussing the responses gives managers the opportunity to explain why some policies are in place and demonstrate how employees should respond in ethical situations they will likely face on the job.
Selecting and Hiring Ethical Employees
Although it is important, or even required in some cases, to have an established and effective code of ethics, it is crucial to the stability of a company to first hire ethical employees. It is better to hire someone who is naturally inclined to behave in an ethical manner than to rely on a company code of ethics to encourage an unethical employee to make ethical choices.
One method of selecting and hiring ethical employees is the use of personality tests or situation-specific questionnaires. Personality tests can be used to determine temperament, outlook (i.e., whether positive or negative), and mood. Situation-specific questionnaires can be customized by a company and used to uncover how a candidate would react when faced with an ethical dilemma.
Calling personal references may also help hiring managers filter out less ethical candidates. Although this method is used often, it is important to be aware that references often withhold negative information about a candidate. Managers should supplement references with another method of weeding out dishonest candidates.
Best practices in the selection and hiring of ethical employees involve including the company’s most ethical employees in the interview process. Interviewers already doing the job a candidate is applying for can ask relevant questions that can lead to answers that reveal a lot about the candidate, such as a propensity to make unethical decisions.
Whistleblower Protection
One of the best enforcement tools is a whistleblower hotline , which is a phone number or other method for employees or other stakeholders to report suspected acts of impropriety, such as fraud, waste, abuse, misconduct, or violations of policy, laws, or regulations. These reports are usually confidential and may be anonymous. Colleagues often see suspicious behavior and are well-positioned to report it or fix it. However, some employees are reluctant to speak up for fear of retaliation. [3]
A whistleblower is an individual such as an employee who reports the misconduct of someone in a position of authority in his or her own organization. In the past, whistleblowers had no protection from retaliation by those they reported—reporting a superior’s wrongdoings often required risking one’s livelihood. This discouraged employees from reporting offenses.
Under the whistleblower protection offered by the Sarbanes-Oxley Act, anyone who reports a corporate wrongdoing and believes he or she could be penalized for it can request that the Occupational Safety and Health Administration (OSHA) investigate the matter. If the whistleblower is an employee who was fired and the termination of employment was found to be improper, the company can be ordered to rehire the individual and give him or her back pay as well as a penalty award.
In 2017, OSHA ordered Wells Fargo to pay $5.4 million to a former manager who said he was fired in 2010 after reporting to his supervisors and to a bank ethics hotline what he suspected was fraudulent behavior. OSHA also said the bank must rehire him.
The $5.4 million, intended to cover back pay, compensatory damages, and legal fees, was the largest individual award ever ordered through OSHA’s whistleblower protection program at the time. [4]
Another agency, the Securities and Exchange Commission, also strengthened whistleblower protections and rewards. In 2011, the SEC issued new whistleblower rules, required by the Dodd-Frank Act. Under the new rules, a whistleblower is eligible for a bounty equal to 10 percent to 30 percent of monetary sanctions resulting from an enforcement action. Accordingly, companies will be faced with the prospect that their employees, suppliers, customers, consultants, advisors, and others may report possible violations directly to the SEC without providing the companies with an opportunity to first investigate the allegations and remediate any violations. Under the act, employers are prohibited from retaliating against whistleblowing employees.
The SEC announced the largest award ever in July 2017. Two whistleblowers are set to share a record $61 million award from the SEC for helping make the case that JPMorgan Chase & Co. failed to disclose to wealthy clients that it was steering them into investments that would be most profitable for the bank. [5]
Whistleblowing pays, and so does an ethics program that prevents the need for whistleblowers.
Fostering Ethical Decisions
In addition to establishing a code of ethics, hiring ethical employees, and protecting whistleblowers, key aspects of senior management’s role in fostering ethical decisions and behavior in any organization include planning, implementing, and communicating the specifics of an ethics program. Ethics programs disclose important corporate values, often through the use of policies and employee/manager training. After a program has been implemented, senior management should monitor and evaluate it and then modify the program as needed or desired to ensure its effectiveness.
It is also important for senior management to model ethical behavior. The responsibilities of senior management in modeling ethical behavior include the following:
- Act ethically and be seen to act ethically.
- Be active in the ethics program. For example, introduce the ethics training or be the person to speak.
- Encourage employees to raise issues.
- Address ethics issues.
- Enforce the ethics program, such as by punishing violators.
Ethics programs are like an insurance policy. They help mitigate the worst consequences when employees do stray, and they help prevent “accidents” by raising employee “safety” awareness.
- In LaToya J. Murray’s “Sarbanes-Oxley Code of Conduct Requirements,” Chron.com, accessed July 29, 2017, http://smallbusiness.chron.com/sarbanes-oxley-code-conduct-requirements-4060.html ↵
- Johnson & Johnson, “Our Credo,” accessed August 03, 2017, https://www.jnj.com/about-jnj/jnj-credo ↵
- Ethics Research Center, “National Business Ethics Survey (NBES) 2013,” accessed August 3, 2017, https://www.ethics.org/ecihome/research/nbes/nbes-reports/nbes-2013 ↵
- Stacy Cowley, “Wells Fargo Whistle-Blower Wins $5.4 Million and His Job Back,” April 3, 2017, https://www.nytimes.com/2017/04/03/business/04-wells-fargo-whistleblower-fired-osha.html?_r=0 . ↵
- Neil Weinberg, “JPMorgan Whistle-Blowers Set to Reap Record $61 Million Bounty,” July 20, 2017, https://www.bloomberg.com/news/articles/2017-07-20/jpmorgan-whistle-blowers-seen-reaping-record-61-million-bounty . ↵
Contributors and Attributions
- How Managers Can Encourage Ethical Behavior. Authored by : Talia Lambarki and Lumen Learning. License : CC BY: Attribution
- Image: The Office of the Whistleblower (SEC) Symbol. Authored by : US Securities and Exchange Commission, The Office of the Whistleblower. Located at : https://commons.wikimedia.org/wiki/File:The_Office_of_the_Whistleblower(SEC)_Symbol.jpg . License : Public Domain: No Known Copyright


2.1.8: Personality Theory in Real Life
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The Application of Frankl’s Theories to the Workplace and Everyday Life
In 1989, Stephen Covey published The 7 Habits of Highly Effective People. Covey’s book became very popular, selling millions of copies on the way to becoming a #1 New York Times bestseller. If you were to read the first chapter of that book now, it would seem very familiar. Covey presents a very existential approach to understanding our lives, particularly with regard to the problems we experience every day. Perhaps it should not be surprising, then, that in the chapters describing the first two of these seven habits he cites and quotes Viktor Frankl numerous times. Indeed, Covey cites Frankl’s first two books as being profoundly influential in his own life, and how impressed Covey was having met Frankl shortly before Frankl’s death (see Covey’s foreword in Pattakos, 2004).
The first two habits, according to Covey, are: 1) be proactive, and 2) begin with the end in mind. He briefly describes Frankl’s experiences in the concentration camps, and refers to Frankl’s most widely quoted saying, that Frankl himself could decide how his experiences would affect him, and that no one could take that freedom away from Frankl! People who choose to develop this level of personal freedom are certainly being proactive, as opposed to responding passively to events that occur around them and to them. It is not necessary, of course, to suffer such tragic circumstances in order to become proactive in one’s own life:
…It is in the ordinary events of every day that we develop the proactive capacity to handle the extraordinary pressures of life. It’s how we make and keep commitments, how we handle a traffic jam, how we respond to an irate customer or a disobedient child. It’s how we view our problems and where we focus our energies. It’s the language we use. (pg. 92; Covey, 1989)
Covey compares his habit of beginning with the end in mind to logotherapy, helping people to recognize the meaning that their life holds. Covey works primarily in business leadership training, so the value of working toward a greater goal than simply keeping a company in business from day to day is clear, especially for those who care about employee morale and quality control (see also Principle-Centered Leadership ; Covey, 1990). When employees share a sense of purpose in their work, they are likely to have higher intrinsic motivation. Think about it for a moment.  Have you ever been given that sort of homework in school or college? So, how much effort did you really put into that job or assignment?
Covey’s remaining habits are: 3) put first things first, 4) think win/win, 5) seek first to understand, then to be understood, 6) synergize, and 7) sharpen the saw. At first glance these principles seem reasonably straight forward, emphasizing practical and responsible actions. However, what does “sharpen the saw” mean? Sharpening the saw refers to keeping our tools in good working order, and we are our most important tool. Covey considers it essential to regularly and consistently, in wise and balanced ways, to exercise the four dimensions of our nature: physical, mental, social/emotional, and spiritual. By investing in ourselves, we are taking care to live an authentic life.
More recently, Covey has examined his principles beyond the business world. In 1997 he published The 7 Habits of Highly Effective Families , a book in which he applies the same 7 habits to family life. Covey certainly has solid credentials as a family man, as father of 9 and grandfather of 43 children, and he won the 2003 Fatherhood Award from the National Fatherhood Initiative. Drawing in large part on his own extensive, personal experience, Covey uses many stories, anecdotes, and examples of real-life situations to help provide context to the challenges of raising a family and how we might best work with them. But first, he introduces a simple process: have a clear vision of what you want to accomplish, have a plan of how you might accomplish it, and use a compass (your own unique gifts that enable you to be an agent of change in your family). In essence, Covey is recommending that you prepare yourself to develop the seven habits. We all know how difficult it is to establish a new habit or break a bad habit; how is your New Year’s resolution going?
Just as families change, so does the world we live in. Recently, Covey addressed this change by proposing an eighth habit (Covey, 2004). He says that this was not simply an important habit he had overlooked before, but one that has risen to new significance as we have fully entered the age of information and technology in the twenty-first century. As communication has become much easier (e.g., email), it has also become less personal and meaningful. Thus the need for the eighth habit: find your voice and inspire others to find theirs. According to Covey, “voice is unique personal significance.” Essentially, it is the same as finding meaning in one’s life, and then helping others to find meaning in their own lives. It is through finding a mission or a purpose in life that we can move “from effectiveness to greatness” (Covey, 2004).
Whereas Covey presented an approach to personal and professional effectiveness (and later to greatness as well) that parallels the principles set forth by Viktor Frankl, Alex Pattakos very directly applies Frankl’s theories to both the workplace and one’s everyday life in Prisoners of Our Thoughts: Viktor Frankl’s Principles at Work (with a foreword by Stephen Covey; Pattakos, 2004). Frankl himself urged Pattakos to publish his book during a meeting in 1996. Pattakos, like Covey, has been profoundly influenced by Frankl’s writings throughout Pattakos’ career. According to Pattakos, we are creatures of habit, and we prefer a life that is both predictable and within our comfort zone. As the world is changing in the twenty-first century, so the conditions under which we work are changing. Pattakos believes there is a need for humanizing work. More than just balancing one’s personal life and career, humanizing work is an attempt to honor our own individuality and to fully engage our human spirit at work. Simply put, it is an effort to apply Frankl’s will-to-meaning in our workplace (Pattakos, 2004).
Like Covey, Pattakos presents seven core principles. They are similar to Covey’s seven habits, but in keeping with Pattakos’ intentions they are aligned more directly with the principles of logotherapy and existential psychology described by Frankl. The seven core principles are: 1) exercise the freedom to choose your attitude, 2) realize your will-to-meaning, 3) detect the meaning of life’s moments, 4) don’t work against yourself, 5) look at yourself from a distance, 6) shift your focus of attention, and 7) extend beyond yourself. These principles include not only the ideas of personal freedom and will-to-meaning, but also dereflection (principles 4 and 6) and the will-to-ultimate-meaning (principle 7). Clearly Pattakos has accomplished his goal of applying logotherapy to the workplace, but how well does this application work in real life?
Pattakos describes the case of a probation officer with the state department of corrections. Rick, as Pattakos identifies him, was raised in foster care and orphanages. However, rather than developing a sense of caring and concern for others who have difficulties in their lives, Rick refers to his clients as “maggots.” Rick has become insensitive and unforgiving, he has also become deeply depressed and anxious. Overall, he feels lost, unhappy, and unfulfilled, and he doesn’t know what to do about it. According to Pattakos, he has become a prisoner of his own thoughts, and only he has the key to his own freedom. Very simply put, he needs to find a new job or find meaning in the one he has now. One possibility is for Rick to consider his own life circumstances in relationship to his clients:
…Whenever we stop long enough to connect to ourselves, to our environment, to those with whom we work, to the task before us, to the extraordinary interdependence that is always part of our lives, we experience meaning. Meaning is who we are in this world. And it is the world that graces us with meaning. (pg. 157; Pattakos, 2004)
By making a responsible choice to seek meaning in our lives, to not work against ourselves, we can put ourselves on a path we had not seen before:
When we live and work with meaning, we can choose to make meaning, to see meaning, and to share meaning. We can choose our attitudes to life and work; we can choose how to respond to others, how to respond to our jobs, and how to make the very best of difficult circumstances. We can transcend ourselves and be transformed by meaning. We can find connection to meaning at work, in the most unusual places and with the most unexpected people. Meaning is full of surprises. (pg. 159; Pattakos, 2004)
And finally, it does not matter what sort of job we have. It is our choice, our freedom:
No matter what our specific job might be, it is the work we do that represents who we are. When we meet our work with enthusiasm, appreciation, generosity, and integrity, we meet it with meaning. And no matter how mundane a job might seem at the time, we can transform it with meaning. Meaning is life’s legacy, and it is as available to us at work as it is available to us in our deepest spiritual quests. We breathe, therefore we are - spiritual. Life is; therefore it is - meaningful. We do, therefore we work.
Viktor Frankl’s legacy was one of hope and possibility. He saw the human condition at its worst, and human beings behaving in ways intolerable to the imagination. He also saw human beings rising to heights of compassion and caring in ways that can only be described as miraculous acts of unselfishness and transcendence. There is something in us that can rise above and beyond everything we think possible… (pg. 162; Pattakos, 2004)
discussion question \(\PageIndex{1}\)
Stephen Covey and Alex Pattakos have applied Frankl’s theories to both the workplace and our everyday lives. How well do you think the principles of existential psychology can address the problems that you face at work, home, school, etc.? Is it ever really as simple as applying one’s will and choosing to act responsibly? Do you live an authentic life?


10.1: Keyword analysis
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In the investigation of relationships between words (or other units of language structure) and texts (or clusters of texts), researchers frequently use a method referred to as keyword analysis . 1 The term was originally used in contexts where cultural values and practices were studied through particular lexical items (cf. Williams 1976, Wierzbicka 2003); in corpus linguistics, it is used in a related but slightly broader sense of words that are characteristic of a particular text, language variety or demographic in the sense that they occur with “unusual frequency in a given text” or set of texts, where “unusual” means high “by comparison with a reference corpus of some kind” (Scott 1997: 236).
In other words, the corpus-linguistic identification of keywords is analogous to the identification of differential collocates, except that it analyses the association of a word W to a particular text (or collection of texts) T in comparison to the language as a whole (as represented by the reference corpus, which is typically a large, balanced corpus). \(Table \text { } 10.1\) shows this schematically.
\(Table \text{ } 10.1\): A generic 2-by-2 table for keyword analysis
Just like collocation analysis, keyword analysis is most often applied inductively, but there is nothing that precludes a deductive design if we have hypotheses about the over- or underrepresentation of particular lexical items in a particular text or collection of texts. In either case, we have two nominal variables: \(\mathrm{Keyword}\) (with the individual words as values) and \(\mathrm{Text}\) (with the values \(\mathrm{text}\) and \(\mathrm{reference \space corpus}\)).
If keyword analysis is applied to a single text, the aim is typically to identify either the topic area or some stylistic property of that text. When applied to text categories, the aim is typically to identify general lexical and/or grammatical properties of the language variety represented by the text categories.
As a first example of the kind of results that keyword analysis yields, consider \(Table \text{ } 10.2\), which shows the 20 most frequent tokens (including punctuation marks) in the LOB corpus and two individual texts (all words were converted to lower case).
As we can see, the differences are relatively small, as all lists are dominated by frequent function words and punctuation marks. Ten of these occur on all three lists ( a , and , in , of , that , the , to , was , the comma and the period), and another six occur on two of them ( as , he , it , on , and opening and closing quotation marks – although the latter are single quotation marks in the case of LOB and double quotation marks in the case of Text B). Even the types that occur only
\(Table \text{ } 10.2\): Most frequent words in three texts (relative frequencies)
once are mostly uninformative with respect to the language variety (or text category) we may be dealing with ( 1959 , at , by , for , had , is , with , the hyphen and opening and closing parentheses). The only exceptions are four content words in Text A: Neosho , river , species , station – these suggest that the text is about the Neosho river and perhaps that it deals with biology (as suggested by the word species).
Applying keyword analysis to each text or collection of texts allows us to identify the words that differ most significantly in frequency from the reference corpus, telling us how the text in question differs lexically from the (written) language of its time as a whole. \(Table \text{ } 10.3\) lists the keywords for Text A.
The keywords now convey a very specific idea of what the text is about: there are two proper names of rivers (the Neosho already seen on the frequency list and the Marais des Cygnes , represented by its constituents Cygnes , Marais and des ), and there are a number of words for specific species of fish as well as the words river and channel .
\(Table \text{ } 10.3\): Keywords in a report on fish populations
The text is clearly about fish in the two rivers. The occurrence of the words station and abundance suggests a research context, which is supported by the occurrence of two dates and opening and closing parentheses (which are often used in scientific texts to introduce references). The text in question is indeed a scientific report on fish populations: Fish Populations , Following a Drought , In the Neosho and Marais des Cygnes Rivers of Kansas (available via Project Gutenberg and in the Supplementary Online Material, file TXQP). Note that the occurrence of some tokens (such as the dates and the parentheses) may be characteristic of a language variety rather than an individual text, a point we will return to below.
Next, consider \(Table \text { } 10.4\), which lists the keywords for Text B. Three things are noticeable: the keyness of a number of words that are most likely proper names
\(Table \text { } 10.4\): Keywords in a science fiction novel
( Hume , Vye , Rynch , Wass , Brodie and Jumala ), pronouns ( he , his ) and punctuation marks indicative of direct speech (the quotation marks and the exclamation mark).
This does not tell us anything about this particular text, but taken together, these pieces of evidence point to a particular genre: narrative text (novels, short stories, etc.). The few potential content words suggest a particular sub-genre: the archaic hunter in combination with the unusual word flitter is suggestive of fantasy or science fiction. If we were to include the next twenty most strongly associated nouns, we would find patrol , camp , needler , safari , guild , tube , planet and out - hunter , which corroborate the impression that we are dealing with a science-fiction novel. And indeed, the text in question is the science-fiction novel 35710 Text Starhunter by Andre Alice Norton (available via Project Gutenberg in the Supplementary Online Material, file TXQP).
Again, the keywords identified are a mixture of topical markers and markers for the language variety (in this case, the genre) of the text, so even a study of the keywords of single texts provides information about more general linguistic properties of the text in question as well as its specific topic. But keyword analysis reveals its true potential when we apply it to clusters of texts, as in the case studies in the next section.
_______________________________
1 The term keyword is frequently spelled as two words ( key word ) or with a hyphen ( key - word ). I have chosen the spelling as a single word here because it seems simplest (at least to me, as a native writer of German, where compounds are always spelled as single words).


4.11: Exercises
Does a cation gain protons to form a positive charge or does it lose electrons?
Iron(III) sulfate [Fe 2 (SO 4 ) 3 ] is composed of Fe 3 + and ions. Explain why a sample of iron(III) sulfate is uncharged.
Which of the following atoms would be expected to form negative ions in binary ionic compounds and which would be expected to form positive ions: P, I, Mg, Cl, In, Cs, O, Pb, Co?
Which of the following atoms would be expected to form negative ions in binary ionic compounds and which would be expected to form positive ions: Br, Ca, Na, N, F, Al, Sn, S, Cd?
Predict the charge on the monatomic ions formed from the following atoms in binary ionic compounds:
(a) P
(b) Mg
(c) Al
(d) O
(e) Cl
(f) Cs
Predict the charge on the monatomic ions formed from the following atoms in binary ionic compounds:
(a) I
(b) Sr
(c) K
(d) N
(e) S
(f) In
Write the electron configuration for each of the following ions:
(a) As 3 –
(b) I –
(c) Be 2 +
(d) Cd 2 +
(e) O 2–
(f) Ga 3 +
(g) Li +
(h) N 3–
(i) Sn 2 +
(j) Co 2 +
(k) Fe 2 +
(l) As 3 +
Write the electron configuration for the monatomic ions formed from the following elements (which form the greatest concentration of monatomic ions in seawater):
(a) Cl
(b) Na
(c) Mg
(d) Ca
(e) K
(f) Br
(g) Sr
(h) F
Write out the full electron configuration for each of the following atoms and for the monatomic ion found in binary ionic compounds containing the element:
(a) Al
(b) Br
(c) Sr
(d) Li
(e) As
(f) S
From the labels of several commercial products, prepare a list of six ionic compounds in the products. For each compound, write the formula. (You may need to look up some formulas in a suitable reference.)
Why is it incorrect to speak of a molecule of solid NaCl?
What information can you use to predict whether a bond between two atoms is covalent or ionic?
Predict which of the following compounds are ionic and which are covalent, based on the location of their constituent atoms in the periodic table:
(a) Cl 2 CO
(b) MnO
(c) NCl 3
(d) CoBr 2
(e) K 2 S
(f) CO
(g) CaF 2
(h) HI
(i) CaO
(j) IBr
(k) CO 2
Explain the difference between a nonpolar covalent bond, a polar covalent bond, and an ionic bond.
From its position in the periodic table, determine which atom in each pair is more electronegative:
(a) Br or Cl
(b) N or O
(c) S or O
(d) P or S
(e) Si or N
(f) Ba or P
(g) N or K
From its position in the periodic table, determine which atom in each pair is more electronegative:
(a) N or P
(b) N or Ge
(c) S or F
(d) Cl or S
(e) H or C
(f) Se or P
(g) C or Si
From their positions in the periodic table, arrange the atoms in each of the following series in order of increasing electronegativity:
(a) C, F, H, N, O
(b) Br, Cl, F, H, I
(c) F, H, O, P, S
(d) Al, H, Na, O, P
(e) Ba, H, N, O, As
From their positions in the periodic table, arrange the atoms in each of the following series in order of increasing electronegativity:
(a) As, H, N, P, Sb
(b) Cl, H, P, S, Si
(c) Br, Cl, Ge, H, Sr
(d) Ca, H, K, N, Si
(e) Cl, Cs, Ge, H, Sr
Which atoms can bond to sulfur so as to produce a positive partial charge on the sulfur atom?
Which is the most polar bond?
(a) C–C
(b) C–H
(c) N–H
(d) O–H
(e) Se–H
Identify the more polar bond in each of the following pairs of bonds:
(a) HF or HCl
(b) NO or CO
(c) SH or OH
(d) PCl or SCl
(e) CH or NH
(f) SO or PO
(g) CN or NN
Which of the following molecules or ions contain polar bonds?
(a) O 3
(b) S 8
(c)
(d)
(e) CO 2
(f) H 2 S
(g)
Name the following compounds:
(a) CsCl
(b) BaO
(c) K 2 S
(d) BeCl 2
(e) HBr
(f) AlF 3
Name the following compounds:
(a) NaF
(b) Rb 2 O
(c) BCl 3
(d) H 2 Se
(e) P 4 O 6
(f) ICl 3
Write the formulas of the following compounds:
(a) rubidium bromide
(b) magnesium selenide
(c) sodium oxide
(d) calcium chloride
(e) hydrogen fluoride
(f) gallium phosphide
(g) aluminum bromide
(h) ammonium sulfate
Write the formulas of the following compounds:
(a) lithium carbonate
(b) sodium perchlorate
(c) barium hydroxide
(d) ammonium carbonate
(e) sulfuric acid
(f) calcium acetate
(g) magnesium phosphate
(h) sodium sulfite
Write the formulas of the following compounds:
(a) chlorine dioxide
(b) dinitrogen tetraoxide
(c) potassium phosphide
(d) silver(I) sulfide
(e) aluminum fluoride trihydrate
(f) silicon dioxide
Write the formulas of the following compounds:
(a) barium chloride
(b) magnesium nitride
(c) sulfur dioxide
(d) nitrogen trichloride
(e) dinitrogen trioxide
(f) tin(IV) chloride
Each of the following compounds contains a metal that can exhibit more than one ionic charge. Name these compounds:
(a) Cr 2 O 3
(b) FeCl 2
(c) CrO 3
(d) TiCl 4
(e) CoCl 2 ∙6H 2 O
(f) MoS 2
Each of the following compounds contains a metal that can exhibit more than one ionic charge. Name these compounds:
(a) NiCO 3
(b) MoO 3
(c) Co(NO 3 ) 2
(d) V 2 O 5
(e) MnO 2
(f) Fe 2 O 3
The following ionic compounds are found in common household products. Write the formulas for each compound:
(a) potassium phosphate
(b) copper(II) sulfate
(c) calcium chloride
(d) titanium(IV) oxide
(e) ammonium nitrate
(f) sodium bisulfate (the common name for sodium hydrogen sulfate)
The following ionic compounds are found in common household products. Name each of the compounds:
(a) Ca(H 2 PO 4 ) 2
(b) FeSO 4
(c) CaCO 3
(d) MgO
(e) NaNO 2
(f) KI
What are the IUPAC names of the following compounds?
(a) manganese dioxide
(b) mercurous chloride (Hg 2 Cl 2 )
(c) ferric nitrate [Fe(NO 3 ) 3 ]
(d) titanium tetrachloride
(e) cupric bromide (CuBr 2 )
Write the Lewis symbols for each of the following ions:
(a) As 3 –
(b) I –
(c) Be 2 +
(d) O 2–
(e) Ga 3+
(f) Li +
(g) N 3–
Many monatomic ions are found in seawater, including the ions formed from the following list of elements. Write the Lewis symbols for the monatomic ions formed from the following elements:
(a) Cl
(b) Na
(c) Mg
(d) Ca
(e) K
(f) Br
(g) Sr
(h) F
Write the Lewis symbols of the ions in each of the following ionic compounds and the Lewis symbols of the atom from which they are formed:
(a) MgS
(b) Al 2 O 3
(c) GaCl 3
(d) K 2 O
(e) Li 3 N
(f) KF
In the Lewis structures listed here, M and X represent various elements in the third period of the periodic table. Write the formula of each compound using the chemical symbols of each element:
(a)
(b)
(c)
(d)
Write the Lewis structure for the diatomic molecule P 2 , an unstable form of phosphorus found in high-temperature phosphorus vapor.
Write Lewis structures for the following:
(a) H 2
(b) HBr
(c) PCl 3
(d) SF 2
(e) H 2 CCH 2
(f) HNNH
(g) H 2 CNH
(h) NO –
(i) N 2
(j) CO
(k) CN –
Write Lewis structures for the following:
(a) O 2
(b) H 2 CO
(c) AsF 3
(d) ClNO
(e) SiCl 4
(f) H 3 O +
(g)
(h)
(i) HCCH
(j) ClCN
(k)
Write Lewis structures for the following:
(a) ClF 3
(b) PCl 5
(c) BF 3
(d)
Write Lewis structures for the following:
(a) SeF 6
(b) XeF 4
(c)
(d) Cl 2 BBCl 2 (contains a B–B bond)
Write Lewis structures for:
(a)
(b)
(c)
(d) HONO
Correct the following statement: “The bonds in solid PbCl 2 are ionic; the bond in a HCl molecule is covalent. Thus, all of the valence electrons in PbCl 2 are located on the Cl – ions, and all of the valence electrons in a HCl molecule are shared between the H and Cl atoms.”
Write Lewis structures for the following molecules or ions:
(a) SbH 3
(b) XeF 2
(c) Se 8 (a cyclic molecule with a ring of eight Se atoms)
Methanol, H 3 COH, is used as the fuel in some race cars. Ethanol, C 2 H 5 OH, is used extensively as motor fuel in Brazil. Both methanol and ethanol produce CO 2 and H 2 O when they burn. Write the chemical equations for these combustion reactions using Lewis structures instead of chemical formulas.
Many planets in our solar system contain organic chemicals including methane (CH 4 ) and traces of ethylene (C 2 H 4 ), ethane (C 2 H 6 ), propyne (H 3 CCCH), and diacetylene (HCCCCH). Write the Lewis structures for each of these molecules.
Carbon tetrachloride was formerly used in fire extinguishers for electrical fires. It is no longer used for this purpose because of the formation of the toxic gas phosgene, Cl 2 CO. Write the Lewis structures for carbon tetrachloride and phosgene.
Identify the atoms that correspond to each of the following electron configurations. Then, write the Lewis symbol for the common ion formed from each atom:
(a) 1 s 2 2 s 2 2 p 5
(b) 1 s 2 2 s 2 2 p 6 3 s 2
(c) 1 s 2 2 s 2 2 p 6 3 s 2 3 p 6 4 s 2 3 d 10
(d) 1 s 2 2 s 2 2 p 6 3 s 2 3 p 6 4 s 2 3 d 10 4 p 4
(e) 1 s 2 2 s 2 2 p 6 3 s 2 3 p 6 4 s 2 3 d 10 4 p 1
The arrangement of atoms in several biologically important molecules is given here. Complete the Lewis structures of these molecules by adding multiple bonds and lone pairs. Do not add any more atoms.
(a) the amino acid serine:
(b) urea:
(c) pyruvic acid:
(d) uracil:
(e) carbonic acid:
A compound with a molar mass of about 28 g/mol contains 85.7% carbon and 14.3% hydrogen by mass. Write the Lewis structure for a molecule of the compound.
A compound with a molar mass of about 42 g/mol contains 85.7% carbon and 14.3% hydrogen by mass. Write the Lewis structure for a molecule of the compound.
Two arrangements of atoms are possible for a compound with a molar mass of about 45 g/mol that contains 52.2% C, 13.1% H, and 34.7% O by mass. Write the Lewis structures for the two molecules.
How are single, double, and triple bonds similar? How do they differ?
Write resonance forms that describe the distribution of electrons in each of these molecules or ions.
(a) selenium dioxide, OSeO
(b) nitrate ion,
(c) nitric acid, HNO 3 (N is bonded to an OH group and two O atoms)
(d) benzene, C 6 H 6 :
(e) the formate ion:
Write resonance forms that describe the distribution of electrons in each of these molecules or ions.
(a) sulfur dioxide, SO 2
(b) carbonate ion,
(c) hydrogen carbonate ion, (C is bonded to an OH group and two O atoms)
(d) pyridine:
(e) the allyl ion:
Write the resonance forms of ozone, O 3 , the component of the upper atmosphere that protects the Earth from ultraviolet radiation.
Sodium nitrite, which has been used to preserve bacon and other meats, is an ionic compound. Write the resonance forms of the nitrite ion,
In terms of the bonds present, explain why acetic acid, CH 3 CO 2 H, contains two distinct types of carbon-oxygen bonds, whereas the acetate ion, formed by loss of a hydrogen ion from acetic acid, only contains one type of carbon-oxygen bond. The skeleton structures of these species are shown:
Write the Lewis structures for the following, and include resonance structures where appropriate. Indicate which has the strongest carbon-oxygen bond.
(a) CO 2
(b) CO
Toothpastes containing sodium hydrogen carbonate (sodium bicarbonate) and hydrogen peroxide are widely used. Write Lewis structures for the hydrogen carbonate ion and hydrogen peroxide molecule, with resonance forms where appropriate.
Determine the formal charge of each element in the following:
(a) HCl
(b) CF 4
(c) PCl 3
(d) PF 5
Determine the formal charge of each element in the following:
(a) H 3 O +
(b)
(c) NH 3
(d)
(e) H 2 O 2
Calculate the formal charge of chlorine in the molecules Cl 2 , BeCl 2 , and ClF 5 .
Calculate the formal charge of each element in the following compounds and ions:
(a) F 2 CO
(b) NO –
(c)
(d)
(e) H 2 CCH 2
(f) ClF 3
(g) SeF 6
(h)
Draw all possible resonance structures for each of these compounds. Determine the formal charge on each atom in each of the resonance structures:
(a) O 3
(b) SO 2
(c)
(d)
Based on formal charge considerations, which of the following would likely be the correct arrangement of atoms in nitrosyl chloride: ClNO or ClON?
Based on formal charge considerations, which of the following would likely be the correct arrangement of atoms in hypochlorous acid: HOCl or OClH?
Based on formal charge considerations, which of the following would likely be the correct arrangement of atoms in sulfur dioxide: OSO or SOO?
Draw the structure of hydroxylamine, H 3 NO, and assign formal charges; look up the structure. Is the actual structure consistent with the formal charges?
Iodine forms a series of fluorides (listed here). Write Lewis structures for each of the four compounds and determine the formal charge of the iodine atom in each molecule:
(a) IF
(b) IF 3
(c) IF 5
(d) IF 7
Write the Lewis structure and chemical formula of the compound with a molar mass of about 70 g/mol that contains 19.7% nitrogen and 80.3% fluorine by mass, and determine the formal charge of the atoms in this compound.
Which of the following structures would we expect for nitrous acid? Determine the formal charges:
Sulfuric acid is the industrial chemical produced in greatest quantity worldwide. About 90 billion pounds are produced each year in the United States alone. Write the Lewis structure for sulfuric acid, H 2 SO 4 , which has two oxygen atoms and two OH groups bonded to the sulfur.
Explain why the HOH molecule is bent, whereas the HBeH molecule is linear.
What feature of a Lewis structure can be used to tell if a molecule’s (or ion’s) electron-pair geometry and molecular structure will be identical?
Explain the difference between electron-pair geometry and molecular structure.
Why is the H–N–H angle in NH 3 smaller than the H–C–H bond angle in CH 4 ? Why is the H–N–H angle in identical to the H–C–H bond angle in CH 4 ?
Explain how a molecule that contains polar bonds can be nonpolar.
As a general rule, MX n molecules (where M represents a central atom and X represents terminal atoms; n = 2 – 5) are polar if there is one or more lone pairs of electrons on M. NH 3 (M = N, X = H, n = 3) is an example. There are two molecular structures with lone pairs that are exceptions to this rule. What are they?
Predict the electron pair geometry and the molecular structure of each of the following molecules or ions:
(a) SF 6
(b) PCl 5
(c) BeH 2
(d)
Identify the electron pair geometry and the molecular structure of each of the following molecules or ions:
(a)
(b) CF 4
(c) BF 3
(d)
(e) BeCl 2
What are the electron-pair geometry and the molecular structure of each of the following molecules or ions?
(a) ClF 5
(b)
(c)
(d) PCl 3
(e) SeF 4
(f)
Predict the electron pair geometry and the molecular structure of each of the following ions:
(a) H 3 O +
(b)
(c)
(d)
(e) ICl 3
(f) XeF 4
(g) SF 2
Identify the electron pair geometry and the molecular structure of each of the following molecules:
(a) ClNO (N is the central atom)
(b) CS 2
(c) Cl 2 CO (C is the central atom)
(d) Cl 2 SO (S is the central atom)
(e) SO 2 F 2 (S is the central atom)
(f) XeO 2 F 2 (Xe is the central atom)
(g) (Cl is the central atom)
Predict the electron pair geometry and the molecular structure of each of the following:
(a) IOF 5 (I is the central atom)
(b) POCl 3 (P is the central atom)
(c) Cl 2 SeO (Se is the central atom)
(d) ClSO + (S is the central atom)
(e) F 2 SO (S is the central atom)
(f)
(g)
Which of the following molecules and ions contain polar bonds? Which of these molecules and ions have dipole moments?
(a) ClF 5
(b)
(c)
(d) PCl 3
(e) SeF 4
(f)
(g) XeF 2
Which of these molecules and ions contain polar bonds? Which of these molecules and ions have dipole moments?
(a) H 3 O +
(b)
(c)
(d)
(e) ICl 3
(f) XeF 4
(g) SF 2
Which of the following molecules have dipole moments?
(a) CS 2
(b) SeS 2
(c) CCl 2 F 2
(d) PCl 3 (P is the central atom)
(e) ClNO (N is the central atom)
Identify the molecules with a dipole moment:
(a) SF 4
(b) CF 4
(c) Cl 2 CCBr 2
(d) CH 3 Cl
(e) H 2 CO
The molecule XF 3 has a dipole moment. Is X boron or phosphorus?
The molecule XCl 2 has a dipole moment. Is X beryllium or sulfur?
Is the Cl 2 BBCl 2 molecule polar or nonpolar?
There are three possible structures for PCl 2 F 3 with phosphorus as the central atom. Draw them and discuss how measurements of dipole moments could help distinguish among them.
Describe the molecular structure around the indicated atom or atoms:
(a) the sulfur atom in sulfuric acid, H 2 SO 4 [(HO) 2 SO 2 ]
(b) the chlorine atom in chloric acid, HClO 3 [HOClO 2 ]
(c) the oxygen atom in hydrogen peroxide, HOOH
(d) the nitrogen atom in nitric acid, HNO 3 [HONO 2 ]
(e) the oxygen atom in the OH group in nitric acid, HNO 3 [HONO 2 ]
(f) the central oxygen atom in the ozone molecule, O 3
(g) each of the carbon atoms in propyne, CH 3 CCH
(h) the carbon atom in Freon, CCl 2 F 2
(i) each of the carbon atoms in allene, H 2 CCCH 2
Draw the Lewis structures and predict the shape of each compound or ion:
(a) CO 2
(b)
(c) SO 3
(d)
A molecule with the formula AB 2 , in which A and B represent different atoms, could have one of three different shapes. Sketch and name the three different shapes that this molecule might have. Give an example of a molecule or ion for each shape.
A molecule with the formula AB 3 , in which A and B represent different atoms, could have one of three different shapes. Sketch and name the three different shapes that this molecule might have. Give an example of a molecule or ion that has each shape.
Draw the Lewis electron dot structures for these molecules, including resonance structures where appropriate:
(a)
(b) CS 2
(c) CS
(d) predict the molecular shapes for and CS 2 and explain how you arrived at your predictions
What is the molecular structure of the stable form of FNO 2 ? (N is the central atom.)
A compound with a molar mass of about 42 g/mol contains 85.7% carbon and 14.3% hydrogen. What is its molecular structure?
Use the simulation to perform the following exercises for a two-atom molecule:
(a) Adjust the electronegativity value so the bond dipole is pointing toward B. Then determine what the electronegativity values must be to switch the dipole so that it points toward A.
(b) With a partial positive charge on A, turn on the electric field and describe what happens.
(c) With a small partial negative charge on A, turn on the electric field and describe what happens.
(d) Reset all, and then with a large partial negative charge on A, turn on the electric field and describe what happens.
Use the simulation to perform the following exercises for a real molecule. You may need to rotate the molecules in three dimensions to see certain dipoles.
(a) Sketch the bond dipoles and molecular dipole (if any) for O 3. Explain your observations.
(b) Look at the bond dipoles for NH 3 . Use these dipoles to predict whether N or H is more electronegative.
(c) Predict whether there should be a molecular dipole for NH 3 and, if so, in which direction it will point. Check the molecular dipole box to test your hypothesis.
Use the Molecule Shape simulator to build a molecule. Starting with the central atom, click on the double bond to add one double bond. Then add one single bond and one lone pair. Rotate the molecule to observe the complete geometry. Name the electron group geometry and molecular structure and predict the bond angle. Then click the check boxes at the bottom and right of the simulator to check your answers.
Use the Molecule Shape simulator to explore real molecules. On the Real Molecules tab, select H 2 O. Switch between the “real” and “model” modes. Explain the difference observed.
Use the Molecule Shape simulator to explore real molecules. On the Real Molecules tab, select “model” mode and S 2 O. What is the model bond angle? Explain whether the “real” bond angle should be larger or smaller than the ideal model angle.


7.6: Concluding Comments: Effects of Imperfect Competition on Economic Welfare
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In imperfectly competitive markets like monopoly, oligopoly, and monopolistic competition, it is often the case that sellers can increase profits at the expense of consumers through their ability to control quantity or price. This is good for the seller but is bad for society as a whole. To conclude this chapter, it is worthwhile to spend some time on the welfare implications of imperfect competition. When markets are imperfectly competitive, the results are that (1) too little is produced; (2) too high of a price is charged for what is produced; and (3) there is a resulting dead-weight loss to the economy. In short, imperfect competition costs the economy in terms of a misallocation of resources. The combined value of economic activity measured as the sum of producers’ and consumers’ surplus would be higher if imperfectly competitive markets behaved competitively. Table \(\PageIndex{1}\) illustrates this point using the Cournot Nash equilibrium example above. Notice that total surplus in the market increases as the market becomes more competitive. Producers’ surplus declines but the gains in consumers’ surplus more than offset the decline in producers’ surplus.
Table \(\PageIndex{1}\). Economic Welfare at Cournot Nash Equilibrium for Different Numbers of Firms*
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Number of Firms
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Producers’ Surplus (Profits)
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Consumers’ Surplus
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Total Surplus
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1 (monopoly)
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$4,050
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$2,025
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$6,075
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2 (oligopoly)
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$3,600
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$3,600
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$7,200
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5 (oligopoly)
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$2,250
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$5,625
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$7,875
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10 (oligopoly)
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$1,339
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$6,694
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$8,033
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Many (approaching perfect competition)
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$32
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$8,068
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$8,100
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*In Table \(\PageIndex{3}\), the market inverse demand curve is \(P = 200 - \Sigma Q_{i}\) and \(MC = 20\) for all firms.
For the simple reason that total surplus increases with stronger competition, most governments have laws in place to encourage competition in markets. These are referred to as antitrust laws. Regulatory agencies evaluate mergers and acquisitions to determine whether they will adversely impact competition. A good example is provided by Hayenga and Wisner (2000). These authors describe antitrust issues that were evaluated in approving a merger between Cargill and Continental Grain, two entities with large grain merchandising businesses. Moreover, antitrust laws make it illegal to fix prices or form a cartel to limit quantity. A famous case of price fixing in agriculture involved lysine, an amino acid used in livestock and poultry feeds. ADM, a US company, conspired with Japanese and South Korean lysine manufacturers to keep the price of lysine above competitive levels. The conspirators faced criminal and civil penalties as a result of the price-fixing conspiracy. A very good overview of this case is presented by Connor (1997). In fact, there was even a movie made about this incident starring Matt Damon.
See the trailer on youtube.com
. The movie is titled
The Informant
and is based on a book by the same name. I am unable to recommend the movie one way or the other as I have not seen it, but it may be something you want to check out some weekend. If you do watch it, I will be interested to hear what you think. Both the Hayenga and Wisner (1997) and Connor (1997) articles are available through the University of Arkansas Libraries.


6.6: Exponential and Logarithmic Equations
- Use like bases to solve exponential equations.
- Use logarithms to solve exponential equations.
- Use the definition of a logarithm to solve logarithmic equations.
- Use the one-to-one property of logarithms to solve logarithmic equations.
- Solve applied problems involving exponential and logarithmic equations.
In 1859, an Australian landowner named Thomas Austin released \(24\) rabbits into the wild for hunting. Because Australia had few predators and ample food, the rabbit population exploded. In fewer than ten years, the rabbit population numbered in the millions.
Uncontrolled population growth, as in the wild rabbits in Australia, can be modeled with exponential functions. Equations resulting from those exponential functions can be solved to analyze and make predictions about exponential growth. In this section, we will learn techniques for solving exponential functions.
Using Like Bases to Solve Exponential Equations
The first technique involves two functions with like bases. Recall that the one-to-one property of exponential functions tells us that, for any real numbers \(b\), \(S\), and \(T\), where \(b>0\), \(b≠1\), \(b^S=b^T\) if and only if \(S=T\).
In other words, when an exponential equation has the same base on each side, the exponents must be equal. This also applies when the exponents are algebraic expressions. Therefore, we can solve many exponential equations by using the rules of exponents to rewrite each side as a power with the same base. Then, we use the fact that exponential functions are one-to-one to set the exponents equal to one another, and solve for the unknown.
For example, consider the equation \(3^{4x−7}=\dfrac{3^{2x}}{3}\). To solve for \(x\), we use the division property of exponents to rewrite the right side so that both sides have the common base, \(3\). Then we apply the one-to-one property of exponents by setting the exponents equal to one another and solving for \(x\):
\[\begin{align*} 3^{4x-7}&= \dfrac{3^{2x}}{3}\\ 3^{4x-7}&= \dfrac{3^{2x}}{3^1} \qquad &&\text{Rewrite 3 as } 3^1\\ 3^{4x-7}&= 3^{2x-1} \qquad &&\text{Use the division property of exponents}\\ 4x-7&= 2x-1 \qquad &&\text{Apply the one-to-one property of exponents}\\ 2x&= 6 \qquad &&\text{Subtract 2x and add 7 to both sides}\\ x&= 3 \qquad &&\text{Divide by 3} \end{align*}\]
For any algebraic expressions \(S\) and \(T\), and any positive real number \(b≠1\),
\[\begin{align} b^S=b^T\text{ if and only if } S=T \end{align}\]
- Use the rules of exponents to simplify, if necessary, so that the resulting equation has the form \(b^S=b^T\).
- Use the one-to-one property to set the exponents equal.
- Solve the resulting equation, \(S=T\), for the unknown.
Solve \(2^{x−1}=2^{2x−4}\).
Solution
\[\begin{align*} 2^{x-1}&= 2^{2x-4} \qquad &&\text{The common base is 2}\\ x-1&= 2x-4 \qquad &&\text{By the one-to-one property the exponents must be equal}\\ x&= 3 \qquad &&\text{Solve for x} \end{align*}\]
Solve \(5^{2x}=5^{3x+2}\).
- Answer
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\(x=−2\)
Rewrite Equations So All Powers Have the Same Base
Sometimes the common base for an exponential equation is not explicitly shown. In these cases, we simply rewrite the terms in the equation as powers with a common base, and solve using the one-to-one property.
For example, consider the equation \(256=4^{x−5}\). We can rewrite both sides of this equation as a power of \(2\). Then we apply the rules of exponents, along with the one-to-one property, to solve for \(x\):
\[\begin{align*} 256&= 4^{x-5}\\ 2^8&= {(2^2)}^{x-5} \qquad &&\text{Rewrite each side as a power with base 2}\\ 2^8&= 2^{2x-10} \qquad &&\text{Use the one-to-one property of exponents}\\ 8&= 2x-10 \qquad &&\text{Apply the one-to-one property of exponents}\\ 18&= 2x \qquad &&\text{Add 10 to both sides}\\ x&= 9 \qquad &&\text{Divide by 2} \end{align*}\]
- Rewrite each side in the equation as a power with a common base.
- Use the rules of exponents to simplify, if necessary, so that the resulting equation has the form \(b^S=b^T\).
- Use the one-to-one property to set the exponents equal.
- Solve the resulting equation, \(S=T\), for the unknown.
Solve \(8^{x+2}={16}^{x+1}\).
Solution
\[\begin{align*} 8^{x+2}&= {16}^{x+1}\\ {(2^3)}^{x+2}&= {(2^4)}^{x+1} \qquad &&\text{Write 8 and 16 as powers of 2}\\ 2^{3x+6}&= 2^{4x+4} \qquad &&\text{To take a power of a power, multiply exponents}\\ 3x+6&= 4x+4 \qquad &&\text{Use the one-to-one property to set the exponents equal}\\ x&= 2 \qquad &&\text{Solve for x} \end{align*}\]
Solve \(5^{2x}={25}^{3x+2}\).
- Answer
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\(x=−1\)
Solve \(2^{5x}=\sqrt{2}\).
Solution
\[\begin{align*} 2^{5x}&= 2^{\frac{1}{2}} \qquad &&\text{Write the square root of 2 as a power of 2}\\ 5x&= \dfrac{1}{2} \qquad &&\text{Use the one-to-one property}\\ x&= \dfrac{1}{10} \qquad &&\text{Solve for x} \end{align*}\]
Solve \(5^x=\sqrt{5}\).
- Answer
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\(x=\dfrac{1}{2}\)
No. Recall that the range of an exponential function is always positive. While solving the equation, we may obtain an expression that is undefined.
Solve \(3^{x+1}=−2\).
Solution
This equation has no solution. There is no real value of \(x\) that will make the equation a true statement because any power of a positive number is positive.
Analysis
Figure \(\PageIndex{2}\) shows that the two graphs do not cross so the left side is never equal to the right side. Thus the equation has no solution.
Solve \(2^x=−100\).
- Answer
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The equation has no solution.
Solving Exponential Equations Using Logarithms
Sometimes the terms of an exponential equation cannot be rewritten with a common base. In these cases, we solve by taking the logarithm of each side. Recall, since \(\log(a)=\log(b)\) is equivalent to \(a=b\), we may apply logarithms with the same base on both sides of an exponential equation.
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Apply the logarithm of both sides of the equation.
- If one of the terms in the equation has base 10, use the common logarithm.
- If none of the terms in the equation has base 10, use the natural logarithm.
- Use the rules of logarithms to solve for the unknown.
Solve \(5^{x+2}=4^x\).
Solution
\[\begin{align*}
5^{x+2}&= 4^x \qquad &&\text{There is no easy way to get the powers to have the same base}\\
\ln5^{x+2}&= \ln4^x \qquad &&\text{Take ln of both sides}\\
(x+2)\ln5&= x\ln4 \qquad &&\text{Use laws of logs}\\
x\ln5+2\ln5&= x\ln4 \qquad &&\text{Use the distributive law}\\
x\ln5-x\ln4&= -2\ln5 \qquad &&\text{Get terms containing x on one side, terms without x on the other}\\
x(\ln5-\ln4)&= -2\ln5 \qquad &&\text{On the left hand side, factor out an x}\\
x\ln \left (\dfrac{5}{4} \right )&= \ln \left (\dfrac{1}{25} \right ) \qquad &&\text{Use the laws of logs}\\
x&=\dfrac{\ln \left (\dfrac{1}{25} \right )}{\ln \left (\dfrac{5}{4} \right )} \qquad &&\text{Divide by the coefficient of x}
\end{align*}\]
Solve \(2^x=3^{x+1}\).
- Answer
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\(x=\dfrac{\ln3}{\ln \left (\dfrac{2}{3} \right )}\)
Yes. The solution is \(0\).
Equations Containing \(e\)
One common type of exponential equations are those with base \(e\). This constant occurs again and again in nature, in mathematics, in science, in engineering, and in finance. When we have an equation with a base \(e\) on either side, we can use the natural logarithm to solve it.
- Divide both sides of the equation by \(A\).
- Apply the natural logarithm of both sides of the equation.
- Divide both sides of the equation by \(k\).
Solve \(100=20e^{2t}\).
Solution
\[\begin{align*} 100&= 20e^{2t}\\ 5&= e^{2t} \qquad &&\text{Divide by the coefficient of the power}\\ \ln5&= 2t \qquad &&\text{Take ln of both sides. Use the fact that } ln(x) \text{ and } e^x \text{ are inverse functions}\\ t&= \dfrac{\ln5}{2} \qquad &&\text{Divide by the coefficient of t} \end{align*}\]
Analysis
Using laws of logs, we can also write this answer in the form \(t=\ln\sqrt{5}\). If we want a decimal approximation of the answer, we use a calculator.
Solve \(3e^{0.5t}=11\).
- Answer
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\(t=2\ln \left (\dfrac{11}{3} \right )\) or \(\ln{ \left (\dfrac{11}{3} \right )}^2\)
No. There is a solution when \(k≠0\),and when \(y\) and \(A\) are either both 0 or neither 0, and they have the same sign. An example of an equation with this form that has no solution is \(2=−3e^t\).
Solve \(4e^{2x}+5=12\).
Solution
\[\begin{align*} 4e^{2x}+5&= 12\\ 4e^{2x}&= 7 \qquad &&\text{Combine like terms}\\ e^{2x}&= \dfrac{7}{4} \qquad &&\text{Divide by the coefficient of the power}\\ 2x&= \ln \left (\dfrac{7}{4} \right ) \qquad &&\text{Take ln of both sides}\\ x&= \dfrac{1}{2}\ln \left (\dfrac{7}{4} \right ) \qquad &&\text{Solve for x} \end{align*}\]
Solve \(3+e^{2t}=7e^{2t}\).
- Answer
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\(t=\ln \left (\dfrac{1}{\sqrt{2}} \right )=−\dfrac{1}{2}\ln(2)\)
Extraneous Solutions
Sometimes the methods used to solve an equation introduce an extraneous solution , which is a solution that is correct algebraically but does not satisfy the conditions of the original equation. One such situation arises in solving when the logarithm is taken on both sides of the equation. In such cases, remember that the argument of the logarithm must be positive. If the number we are evaluating in a logarithm function is negative, there is no output.
Solve \(e^{2x}−e^x=56\).
Solution
\[\begin{align*} e^{2x}-e^x&= 56\\ e^{2x}-e^x-56&= 0 \qquad &&\text{Get one side of the equation equal to zero}\\ (e^x+7)(e^x-8)&= 0 \qquad &&\text{Factor by the FOIL method}\\ e^x+7&= 0 \qquad &&\text{or} \\ e^x-8&= 0 \qquad &&\text{If a product is zero, then one factor must be zero}\\ e^x&= -7 \qquad &&\text{or} \\ e^x&= 8 \qquad &&\text{Isolate the exponentials}\\ e^x&= 8 \qquad &&\text{Reject the equation in which the power equals a negative number}\\ x&= \ln8 \qquad &&\text{Solve the equation in which the power equals a positive number} \end{align*}\]
Analysis
When we plan to use factoring to solve a problem, we always get zero on one side of the equation, because zero has the unique property that when a product is zero, one or both of the factors must be zero. We reject the equation \(e^x=−7\) because a positive number never equals a negative number. The solution \(\ln(−7)\) is not a real number, and in the real number system this solution is rejected as an extraneous solution.
Solve \(e^{2x}=e^x+2\).
- Answer
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\(x=\ln2\)
No. Keep in mind that we can only apply the logarithm to a positive number. Always check for extraneous solutions.
Using the Definition of a Logarithm to Solve Logarithmic Equations
We have already seen that every logarithmic equation \({\log}_b(x)=y\) is equivalent to the exponential equation \(b^y=x\). We can use this fact, along with the rules of logarithms, to solve logarithmic equations where the argument is an algebraic expression.
For example, consider the equation \({\log}_2(2)+{\log}_2(3x−5)=3\). To solve this equation, we can use rules of logarithms to rewrite the left side in compact form and then apply the definition of logs to solve for \(x\):
\[\begin{align*} {\log}_2(2)+{\log}_2(3x-5)&= 3\\ {\log}_2(2(3x-5))&= 3 \qquad \text{Apply the product rule of logarithms}\\ {\log}_2(6x-10)&= 3 \qquad \text{Distribute}\\ 2^3&= 6x-10 \qquad \text{Apply the definition of a logarithm}\\ 8&= 6x-10 \qquad \text{Calculate } 2^3\\ 18&= 6x \qquad \text{Add 10 to both sides}\\ x&= 3 \qquad \text{Divide by 6} \end{align*}\]
For any algebraic expression \(S\) and real numbers \(b\) and \(c\),where \(b>0\), \(b≠1\),
\[\begin{align} {\log}_b(S)=c \text{ if and only if } b^c=S \end{align}\]
Solve \(2\ln x+3=7\).
Solution
\[\begin{align*} 2\ln x+3&= 7\\ 2\ln x&= 4 \qquad \text{Subtract 3}\\ \ln x&= 2 \qquad \text{Divide by 2}\\ x&= e^2 \qquad \text{Rewrite in exponential form} \end{align*}\]
Solve \(6+\ln x=10\).
- Answer
-
\(x=e^4\)
Solve \(2\ln(6x)=7\).
Solution
\[\begin{align*} 2\ln(6x)&= 7\\ \ln(6x)&= \dfrac{7}{2} \qquad \text{Divide by 2}\\ 6x&= e^{\left (\dfrac{7}{2} \right )} \qquad \text{Use the definition of }\ln \\ x&= \dfrac{1}{6}e^{\left (\dfrac{7}{2} \right )} \qquad \text{Divide by 6} \end{align*}\]
Solve \(2\ln(x+1)=10\).
- Answer
-
\(x=e^5−1\)
Solve \(\ln x=3\).
Solution
\[\begin{align*} \ln x&= 3\\ x&= e^3 \qquad \text{Use the definition of the natural logarithm} \end{align*}\]
Figure \(\PageIndex{3}\) represents the graph of the equation. On the graph, the x -coordinate of the point at which the two graphs intersect is close to \(20\). In other words \(e^3≈20\). A calculator gives a better approximation: \(e^3≈20.0855\).
Use a graphing calculator to estimate the approximate solution to the logarithmic equation \(2^x=1000\) to \(2\) decimal places.
- Answer
-
\(x≈9.97\)
Using the One-to-One Property of Logarithms to Solve Logarithmic Equations
As with exponential equations, we can use the one-to-one property to solve logarithmic equations. The one-to-one property of logarithmic functions tells us that, for any real numbers \(x>0\), \(S>0\), \(T>0\) and any positive real number \(b\), where \(b≠1\),
\({\log}_bS={\log}_bT\) if and only if \(S=T\).
For example,
If \({\log}_2(x−1)={\log}_2(8)\), then \(x−1=8\).
So, if \(x−1=8\), then we can solve for \(x\),and we get \(x=9\). To check, we can substitute \(x=9\) into the original equation: \({\log}_2(9−1)={\log}_2(8)=3\). In other words, when a logarithmic equation has the same base on each side, the arguments must be equal. This also applies when the arguments are algebraic expressions. Therefore, when given an equation with logs of the same base on each side, we can use rules of logarithms to rewrite each side as a single logarithm. Then we use the fact that logarithmic functions are one-to-one to set the arguments equal to one another and solve for the unknown.
For example, consider the equation \(\log(3x−2)−\log(2)=\log(x+4)\). To solve this equation, we can use the rules of logarithms to rewrite the left side as a single logarithm, and then apply the one-to-one property to solve for \(x\):
\[\begin{align*} \log(3x-2)-\log(2)&= \log(x+4)\\ \log \left (\dfrac{3x-2}{2} \right )&= \log(x+4) \qquad \text{Apply the quotient rule of logarithms}\\ \dfrac{3x-2}{2}&= x+4 \qquad \text{Apply the one to one property of a logarithm}\\ 3x-2&= 2x+8 \qquad \text{Multiply both sides of the equation by 2}\\ x&= 10 \qquad \text{Subtract 2x and add 2} \end{align*}\]
To check the result, substitute \(x=10\) into \(\log(3x−2)−\log(2)=\log(x+4)\).
\[\begin{align*} \log(3(10)-2)-\log(2)&= \log((10)+4) \\ \log(28)-\log(2)&= \log(14)\\ \log \left (\dfrac{28}{2} \right )&= \log(14) \qquad \text{The solution checks} \end{align*}\]
For any algebraic expressions \(S\) and \(T\) and any positive real number \(b\), where \(b≠1\),
\[\begin{align} b^S=b^T\text{ if and only if } S=T \end{align}\]
Note, when solving an equation involving logarithms, always check to see if the answer is correct or if it is an extraneous solution.
- Use the rules of logarithms to combine like terms, if necessary, so that the resulting equation has the form \({\log}_bS={\log}_bT\).
- Use the one-to-one property to set the arguments equal.
- Solve the resulting equation, \(S=T\), for the unknown.
Solve \(\ln(x^2)=\ln(2x+3)\).
Solution
\[\begin{align*} \ln(x^2)&= \ln(2x+3)\\ x^2&= 2x+3 \qquad \text{Use the one-to-one property of the logarithm}\\ x^2-2x-3&= 0 \qquad \text{Get zero on one side before factoring}\\ (x-3)(x+1)&= 0 \qquad \text{Factor using FOIL}\\ x-3&= 0 \qquad \text{or } x+1=0 \text{ If a product is zero, one of the factors must be zero}\\ x=3 \qquad \text{or} \\ x&= -11 \qquad \text{Solve for x} \end{align*}\]
Analysis
There are two solutions: \(3\) or \(−1\). The solution \(−1\) is negative, but it checks when substituted into the original equation because the argument of the logarithm functions is still positive.
Solve \(\ln(x^2)=\ln1\).
- Answer
-
\(x=1\) or \(x=−1\)
Solving Applied Problems Using Exponential and Logarithmic Equations
In previous sections, we learned the properties and rules for both exponential and logarithmic functions. We have seen that any exponential function can be written as a logarithmic function and vice versa. We have used exponents to solve logarithmic equations and logarithms to solve exponential equations. We are now ready to combine our skills to solve equations that model real-world situations, whether the unknown is in an exponent or in the argument of a logarithm.
One such application is in science, in calculating the time it takes for half of the unstable material in a sample of a radioactive substance to decay, called its half-life . Table \(\PageIndex{1}\) lists the half-life for several of the more common radioactive substances.
| Substance | Use | Half-life |
|---|---|---|
| gallium-67 | nuclear medicine | 80 hours |
| cobalt-60 | manufacturing | 5.3 years |
| technetium-99m | nuclear medicine | 6 hours |
| americium-241 | construction | 432 years |
| carbon-14 | archeological dating | 5,715 years |
| uranium-235 | atomic power | 703,800,000 years |
We can see how widely the half-lives for these substances vary. Knowing the half-life of a substance allows us to calculate the amount remaining after a specified time. We can use the formula for radioactive decay:
\[\begin{align} A(t)&= A_0e^{\tfrac{\ln(0.5)}{T}t}\\ A(t)&= A_0e^{\tfrac{\ln(0.5)t}{T}}\\ A(t)&= A_0{(e^{\ln(0.5)})}^{\tfrac{t}{T}}\\ A(t)&= A_0{\left (\dfrac{1}{2}\right )}^{\tfrac{t}{T}}\\ \end{align}\]
where
- \(A_0\) is the amount initially present
- \(T\) is the half-life of the substance
- \(t\) is the time period over which the substance is studied
- \(y\) is the amount of the substance present after time \(t\)
How long will it take for ten percent of a \(1000\)-gram sample of uranium-235 to decay?
Solution
\[\begin{align*} y&= 1000e^{\tfrac{\ln(0.5)}{703,800,000}t}\\ 900&= 1000e^{\tfrac{\ln(0.5)}{703,800,000}t} \qquad \text{After } 10\% \text{ decays, 900 grams are left}\\ 0.9&= e^{\tfrac{\ln(0.5)}{703,800,000}t} \qquad \text{Divide by 1000}\\ \ln(0.9)&= \ln \left (e^{\tfrac{\ln(0.5)}{703,800,000}t} \right ) \qquad \text{Take ln of both sides}\\ \ln(0.9)&= \dfrac{\ln(0.5)}{703,800,000}t \qquad \ln(e^M)=M\\ t&= 703,800,000\times \dfrac{\ln(0.9)}{\ln(0.5)} \qquad \text{years Solve for t}\\ t&\approx 106,979,777 \qquad \text{years} \end {align*} \]
Analysis
Ten percent of \(1000\) grams is \(100\) grams. If \(100\) grams decay, the amount of uranium-235 remaining is \(900\) grams.
How long will it take before twenty percent of our 1000-gram sample of uranium-235 has decayed?
- Answer
-
\(t=703,800,000×\dfrac{\ln(0.8)}{\ln(0.5)}\)years ≈ 226,572,993 years.
Access these online resources for additional instruction and practice with exponential and logarithmic equations.
- Solving Logarithmic Equations
- Solving Exponential Equations with Logarithms
Key Equations
| One-to-one property for exponential functions | For any algebraic expressions \(S\) and \(T\) and any positive real number \(b\), where \(b^S=b^T\) if and only if \(S=T\). |
| Definition of a logarithm | For any algebraic expression S and positive real numbers \(b\) and \(c\), where \(b≠1\), \({\log}_b(S)=c\) if and only if \(b^c=S\). |
| One-to-one property for logarithmic functions |
For any algebraic expressions \(S\) and \(T\) and any positive real number \(b\), where \(b≠1\),
\({\log}_bS={\log}_bT\) if and only if \(S=T\). |
Key Concepts
- We can solve many exponential equations by using the rules of exponents to rewrite each side as a power with the same base. Then we use the fact that exponential functions are one-to-one to set the exponents equal to one another and solve for the unknown.
- When we are given an exponential equation where the bases are explicitly shown as being equal, set the exponents equal to one another and solve for the unknown. See Example \(\PageIndex{1}\).
- When we are given an exponential equation where the bases are not explicitly shown as being equal, rewrite each side of the equation as powers of the same base, then set the exponents equal to one another and solve for the unknown. See Example \(\PageIndex{2}\), Example \(\PageIndex{3}\), and Example \(\PageIndex{4}\).
- When an exponential equation cannot be rewritten with a common base, solve by taking the logarithm of each side. See Example \(\PageIndex{5}\).
- We can solve exponential equations with base \(e\),by applying the natural logarithm of both sides because exponential and logarithmic functions are inverses of each other. See Example \(\PageIndex{6}\) and Example \(\PageIndex{7}\).
- After solving an exponential equation, check each solution in the original equation to find and eliminate any extraneous solutions. See Example \(\PageIndex{8}\).
- When given an equation of the form \({\log}_b(S)=c\), where \(S\) is an algebraic expression, we can use the definition of a logarithm to rewrite the equation as the equivalent exponential equation \(b^c=S\), and solve for the unknown. See Example \(\PageIndex{9}\) and Example \(\PageIndex{10}\).
- We can also use graphing to solve equations with the form \({\log}_b(S)=c\). We graph both equations \(y={\log}_b(S)\) and \(y=c\) on the same coordinate plane and identify the solution as the x- value of the intersecting point. See Example \(\PageIndex{11}\).
- When given an equation of the form \({\log}_bS={\log}_bT\), where \(S\) and \(T\) are algebraic expressions, we can use the one-to-one property of logarithms to solve the equation \(S=T\) for the unknown. See Example \(\PageIndex{12}\).
- Combining the skills learned in this and previous sections, we can solve equations that model real world situations, whether the unknown is in an exponent or in the argument of a logarithm. See Example \(\PageIndex{13}\).


15: The Sun- A Garden-Variety Star
“Space weather” may sound like a contradiction. How can there be weather in the vacuum of space? Yet space weather, which refers to changing conditions in space, is an active field of research and can have profound effects on Earth. We are all familiar with the ups and downs of weather on Earth, and how powerful storms can be devastating for people and vegetation. Although we are separated from the Sun by a large distance as well as by the vacuum of space, we now understand that great outbursts on the Sun (solar storms, in effect) can cause changes in the atmosphere and magnetic field of Earth, sometimes even causing serious problems on the ground. In this chapter, we will explore the nature of the Sun’s outer layers, the changing conditions and activity there, and the ways that the Sun affects Earth.
By studying the Sun, we also learn much that helps us understand stars in general. The Sun is, in astronomical terms, a rather ordinary star—not unusually hot or cold, old or young, large or small. Indeed, we are lucky that the Sun is typical. Just as studies of Earth help us understand observations of the more distant planets, so too does the Sun serve as a guide to astronomers in interpreting the messages contained in the light we receive from distant stars. As you will learn, the Sun is dynamic, continuously undergoing change, balancing the forces of nature to keep itself in equilibrium. In this chapter, we describe the components of the Sun, how it changes with time, and how those changes affect Earth.
-
- 15.2: The Structure and Composition of the Sun
- The Sun, our star, has several layers beneath the visible surface: the core, radiative zone, and convective zone. These, in turn, are surrounded by a number of layers that make up the solar atmosphere. In order of increasing distance from the center of the Sun, they are the photosphere, with a temperature that ranges from 4500 K to about 6800 K; the chromosphere, with a typical temperature of 104 K; the transition region, a zone that may be only a few kilometers thick.
-
- 15.3: The Solar Cycle
- Sunspots are dark regions where the temperature is up to 2000 K cooler than the surrounding photosphere. Their motion across the Sun’s disk allows us to calculate how fast the Sun turns on its axis. The Sun rotates more rapidly at its equator, where the rotation period is about 25 days, than near the poles, where the period is slightly longer than 36 days. The number of visible sunspots varies according to a sunspot cycle that averages 11 years in length. Spots frequently occur in pairs.
-
- 15.4: Solar Activity above the Photosphere
- Signs of more intense solar activity, an increase in the number of sunspots, as well as prominences, plages, solar flares, and coronal mass ejections, all tend to occur in active regions—that is, in places on the Sun with the same latitude and longitude but at different heights in the atmosphere. Active regions vary with the solar cycle, just like sunspots do.
-
- 15.5: Space Weather
- Space weather is the effect of solar activity on our own planet, both in our magnetosphere and on Earth’s surface. Coronal holes allow more of the Sun’s material to flow out into space. Solar flares and coronal mass ejections can cause auroras, disrupt communications, damage satellites, and cause power outages on Earth.
Thumbnail: The Sun—our local star—is quite average in many ways. However, that does not stop it from being a fascinating object to study. From solar flares and coronal mass ejections, like the one seen coming from the Sun in the top right of this image, the Sun is a highly dynamic body at the center of our solar system. This image combines two separate satellite pictures of the Sun—the inner one from the Solar Dynamics Observatory and the outer one from the Solar and Heliospheric Observatory. (credit: modification of work by ESA/NASA)


1.2: Basic Objects in the Universe
This module starts with the question, “What is Astronomy? ”Simply stated, Astronomy is the study of the Universe. Astronomers classify objects into various groups based on characteristics and locations in our Universe. Some of these classifications, such as stars, are easy to understand scientifically. Other objects are more difficult to classify, and in some instances, to entirely understand. This module identifies basic objects in our Solar System and those beyond, the overall scale of the Universe, and patterns in the night sky.

Planets
Planets are moderately large objects orbiting a star. We see planets because they reflect the light of their central star, or in some cases, stars. Planets are generally rocky or gaseous in nature and spherical-shaped.
A new group of objects has been recently defined: the Dwarf Planets or Plutoids. These are objects that orbit the Sun, but have not cleared their orbits. Pluto is an example of a Dwarf Planet.
Public Domain | Image courtesy of NASA.
Satellite
A satellite orbits a planet; these objects are also called moons. For example, the Earth’s satellite is the Moon – a proper name.
Asteroid
An asteroid is a relatively small, rocky/metallic object usually orbiting a star.
Public Domain | Image courtesy of NASA.
Comet
A comet is a relatively small, icy object usually orbiting a star. Asteroids, comets, and miscellaneous small/irregular objects and “dust” are often categorized as Minor Bodies.
Solar System
The Solar System is the Sun and all the objects that orbit the Sun, including the planets and their moons.
Stellar System
A Stellar System is a star and other objects such as planets and/or other stars and other materials that orbit it.
Galaxy
A galaxy is a large island of stars, a few hundred million to over a trillion stars.
Galactic Cluster
A Galactic Cluster is a collection of galaxies gravitationally bound.
Supercluster
A Supercluster is a region where galaxies and galactic clusters are tightly packed.
Universe
The Universe is all matter and energy, and is also called the Cosmos.


4: Linked Lists
In this chapter, we continue to study implementations of the List interface, this time using pointer-based data structures rather than arrays. The structures in this chapter are made up of nodes that contain the list items. Using references (pointers), the nodes are linked together into a sequence. We first study singly-linked lists, which can implement Stack and (FIFO) Queue operations in constant time per operation and then move on to doubly-linked lists, which can implement Deque operations in constant time.
Linked lists have advantages and disadvantages when compared to array-based implementations of the List interface. The primary disadvantage is that we lose the ability to access any element using \(\mathtt{get(i)}\) or \(\mathtt{set(i,x)}\) in constant time. Instead, we have to walk through the list, one element at a time, until we reach the \(\mathtt{i}\)th element. The primary advantage is that they are more dynamic: Given a reference to any list node \(\mathtt{u}\), we can delete \(\mathtt{u}\) or insert a node adjacent to \(\mathtt{u}\) in constant time. This is true no matter where \(\mathtt{u}\) is in the list.