11.2D: Pathogen Recognition
- Describe the role of PAMPs and PRRs, interferons, and other cytokines in innate immunity
Pathogen recognition
When a pathogen enters the body, cells in the blood and lymph detect the specific pathogen-associated molecular patterns (PAMPs) on the pathogen’s surface. PAMPs are carbohydrate, polypeptide, and nucleic acid “signatures” that are expressed by viruses, bacteria, and parasites, but which differ from molecules on host cells. These PAMPs allow the immune system to recognize “self” from “other” so as not to destroy the host.
The immune system has specific cells with receptors that recognize these PAMPs. A macrophage is a large, phagocytic cell that engulfs foreign particles and pathogens. Macrophages recognize PAMPs via complementary pattern recognition receptors (PRRs). PRRs are molecules on macrophages and dendritic cells which are in contact with the external environment and can thus recognize PAMPs when present. A monocyte, a type of leukocyte (white blood cell) that circulates in the blood and lymph, differentiates into macrophages after it moves into infected tissue. Dendritic cells bind molecular signatures of pathogens, promoting pathogen engulfment and destruction.
Once a pathogen is detected, the immune system must also track whether it is replicating intracellularly (inside the cell, as with most viruses and some bacteria) or extracellularly (outside of the cell, as with other bacteria, but not viruses). The innate immune system must respond accordingly by identifying the extracellular pathogen and/or by identifying host cells that have already been infected.
Cytokine release affect
The binding of PRRs with PAMPs triggers the release of cytokines, which signal that a pathogen is present and needs to be destroyed along with any infected cells. A cytokine is a chemical messenger that regulates cell differentiation (form and function), proliferation (production), and gene expression to affect immune responses. At least 40 types of cytokines exist in humans that differ in terms of the cell type that produces them, the cell type that responds to them, and the changes they produce.
One subclass of cytokines is the interleukin (IL), which mediates interactions between leukocytes (white blood cells). Interleukins are involved in bridging the innate and adaptive immune responses. In addition to being released from cells after PAMP recognition, cytokines are released by the infected cells which bind to nearby uninfected cells, inducing those cells to release cytokines, resulting in a cytokine burst.
A second class of cytokines is interferons, which are released by infected cells as a warning to nearby uninfected cells. A function an interferons is to inhibit viral replication, making them particularly effective against viruses. They also have other important functions, such as tumor surveillance. Interferons work by signaling neighboring uninfected cells to destroy RNA (often a very important biomolecule for viruses) and reduce protein synthesis; signaling neighboring infected cells to undergo apoptosis (programmed cell death); and activating immune cells.
Cytokines also send feedback to cells of the nervous system to bring about the overall symptoms of feeling sick, which include lethargy, muscle pain, and nausea. These effects may have evolved because the symptoms encourage the individual to rest, preventing them from spreading the infection to others. Cytokines also increase the core body temperature, causing a fever, which causes the liver to withhold iron from the blood. Without iron, certain pathogens (such as some bacteria) are unable to replicate; this is called nutritional immunity.
Phagocytosis and inflammation
The first cytokines to be produced are pro-inflammatory; that is, they encourage inflammation, or the localized redness, swelling (edema), heat, loss of function, and pain that result from the movement of leukocytes and fluid through increasingly-permeable capillaries to a site of infection. The population of leukocytes that arrives at an infection site depends on the nature of the infecting pathogen. Both macrophages and dendritic cells engulf pathogens and cellular debris through phagocytosis. A neutrophil is also a phagocytic leukocyte that engulfs and digests pathogens. Neutrophils, the most-abundant leukocytes of the immune system, have a nucleus with two to five lobes and contain organelles (lysosomes) that digest engulfed pathogens. An eosinophil is a leukocyte that works with other eosinophils to surround a parasite. It is involved in the allergic response and in protection against helminthes (parasitic worms).
Neutrophils and eosinophils are particularly important leukocytes that engulf large pathogens, such as bacteria and fungi. A mast cell is a leukocyte that produces inflammatory molecules, such as histamine, in response to large pathogens. A basophil is a leukocyte that, like a neutrophil, releases chemicals to stimulate the inflammatory response. Basophils are also involved in allergy and hypersensitivity responses and induce specific types of inflammatory responses. Eosinophils and basophils produce additional inflammatory mediators to recruit more leukocytes. A hypersensitive immune response to harmless antigens, such as in pollen, often involves the release of histamine by basophils and mast cells; this is why many anti-allergy medications are anti-histamines.
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Key Points
- Pathogens are recognized by a variety of immune cells, such as macrophages and dendritic cells, via pathogen-associated molecular patterns (PAMPs) on the pathogen surface, which interact with complementary pattern-recognition receptors (PRRs) on the immune cells’ surfaces.
- Upon binding of PRRs with PAMPs (pathogen recognition), immune cells release cytokines to tell other cells to start fighting back.
- One class of cytokines, interferons, warn nearby uninfected cells of impending infection, cause cells to start cleaving RNA and reduce protein synthesis, and signal nearby infected cells to undergo apoptosis.
- Another class of cytokines, called inerleukins, mediate interactions between white blood cells ( leukocytes ) and help bridge the innate and adaptive immune responses.
- Inflammation (hot, red, swollen, painful tissue associated with infection) is encouraged by cytokines that are produced immediately upon pathogen recognition; the increase in blood flow associated with inflammation allows more leukocytes (a type of innate immune cell) to reach the infected area.
Key Terms
- macrophage : a white blood cell that phagocytizes necrotic cell debris and foreign material, including viruses, bacteria, and tattoo ink; part of the innate immune system
- phagocytosis : the process where a cell incorporates a particle by extending pseudopodia and drawing the particle into a vacuole of its cytoplasm
- cytokine : any of various small regulatory proteins that regulate the cells of the immune system; they are released upon binding of PRRs to PAMPS


Many of the Indigenous languages spoken in what is currently Canada are quite endangered, because of deliberate strategies by the settler government. For over 100 years, children from Indigenous communities were forced to attend residential schools where they were severely punished for speaking their home languages. The consequence was that fewer and fewer Indigenous people were able to maintain fluency in their languages.
Video Script
A lot of our attention so far has focused on English, which is convenient because it’s a language that we all know, but we can learn a lot about mental grammar by looking at other languages. Canada has an incredibly rich and diverse history of languages that were spoken by Aboriginal peoples long before European settlers arrived. Linguists estimate that there were more than two hundred different Indigenous languages spoken in this region, and these languages were quite different from each other — they formed about 15 different language families.
At the time of the 2016 census in Canada, there were still about two hundred and thirteen thousand people speaking about 64 different Indigenous languages from 12 different language families. Some of these languages, like Cree, Inuktitut and Ojibwe, are quite healthy, with thousands of speakers. But many more Indigenous languages are critically endangered — they have only a few hundred or a few dozen speakers who are quite elderly. When those speakers die, the language could die with them.
Why have so many of the Aboriginal languages been lost? It’s tempting to attribute it to economic and cultural pressures — TV shows and books and music are all in English, and everyone wants to speak English to get a job — but it’s not as simple as that. From the time that European settlers first arrived in this region, they engaged in deliberate strategies to try to eliminate First Nations people and their culture and language. The settlers engaged in war with the Indigenous people and brought new germs that caused devastating epidemics. The Europeans took over fertile land to grow their own crops and forced Indigenous people to live in small, confined reserves that could not sustain the crops to feed their people.
And these strategies aren’t just from hundreds of years ago: between the 1960s and the late 1980s, the Canadian government seized thousands of Aboriginal children from their homes and placed them forcibly in foster homes and adoptive homes largely with white families, which meant that the children did not learn their parents’ language. This forced adoption is sometimes called the “sixties scoop”, and it continued the tradition of the residential schools.
The residential school system existed in Canada for more than 100 years, and the last residential school closed in 1996, not very long ago. Aboriginal children were taken from their families and forced to live in quite appalling conditions in schools that were run by the government and by the churches. The person who initiated the system was Sir John A Macdonald, Canada’s first prime minister. He was quite clear that the whole purpose of taking children from their families was to make sure that they grew up without knowledge of their history, language, and culture. Here’s his attitude about children who grow up in their families and communities:
“When the school is on the reserve the child lives with its parents, who are savages; he is surrounded by savages, and though he may learn to read and write, his habits, and training and mode of thought are Indian. He is simply a savage who can read and write.”
And here’s what his plan was:
“Indian children should be withdrawn as much as possible from the parental influence, and the only way to do that would be to put them in central training industrial schools where they will acquire the habits and modes of thought of white men.”
He was completely open about his goals: he wanted Aboriginal children to stop thinking and speaking in the ways they learned in their families and communities, and to start thinking and speaking like white men.
In 2015, Canada’s Truth and Reconciliation Commission issued its report, after years of consulting with survivors of residential schools. The executive summary begins this way:
“These residential schools were created for the purpose of separating Aboriginal children from their families, in order to minimize and weaken family ties and cultural linkages, and to indoctrinate children into a new culture—the culture of the legally dominant Euro-Christian Canadian society.”
The TRC’s Calls to Action acknowledge the crucial role that Indigenous languages will play in achieving reconciliation between Aboriginal people and the larger Canadian population. Here are just some of the Calls to Action.
-
We call on the federal government to draft new Aboriginal education legislation including protecting the right to Aboriginal languages [and] the teaching of Aboriginal languages as credit courses.
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We call upon the federal government to enact an Aboriginal Languages Act that incorporates the following principles:
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Aboriginal languages are a fundamental and valued element of Canadian culture and society, and there is an urgency to preserve them.
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The federal government has a responsibility to provide sufficient funds for Aboriginal-language revitalization and preservation.
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The preservation, revitalization, and strengthening of Aboriginal languages and cultures are best managed by Aboriginal people and communities.
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Funding for Aboriginal language initiatives must reflect the diversity of Aboriginal languages.
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We call upon post-secondary institutions to create university and college degree and diploma programs in Aboriginal languages.
So there is a need for language preservation, revitalization and teaching. What can linguists do to help with these efforts? Of course the most important thing is to work with Aboriginal communities, to listen to the community members and find out from them what they think would be most valuable.
Some linguists have helped to document Indigenous languages, recording and transcribing speech and stories from native speakers. This is especially crucial when the speakers are elderly and the language is critically endangered. Documenting a language also involves doing phonological, morphological and syntactic analysis to be able to write grammar books and dictionaries for the language. Some linguists have also helped to develop writing systems for languages that didn’t have any written form. If a language has been documented, then linguists can help to create educational resources and curriculum material that language teachers can use, and can help to train language teachers.


3.1.2: Shape (Plane)
Shape is an area that stands out from the space next to or around it due to a defined or implied boundary often due to differences in value, texture, or color.
Just like line, shapes have a variety of possible characteristics. One of the most basic is the difference between being geometric or organic in nature. Geometric 2D shapes are typically made by humans and are made up of circles, squares, and triangles. Geometric shapes are derived from mathematics and can be curvilinear (made from curves) or rectilinear (made of straight lines).
In contrast, organic or biomorphic shapes are those generally found in nature. These shapes often have rounded, curving forms suggesting natural processes like growth, erosion, or decay. Humans are organic shapes and have worked with natural materials to create new forms over the history of our species.
Tiffany was looking to create a very organic form in both the shape, color, and pattern in the surface of this vase.
Another important characteristic when discussing shape is the difference between mass and volume. In 2D design, mass and volume are implied, the illusion made by using shading to add value. Mass is defined as the physical weight of an object. Stone, clay, plaster, concrete, and metal objects generally appear to have a large mass.
These pieces were constructed hollow in order to survive the kiln firing process but still have considerable visual mass.
Volume is how much space an object occupies. A massive shape generally takes up a large amount of space, but a lighter object can also have the same volume and occupy the same amount of space. This concept can be simplified to: All mass has volume, but not all volume has mass.


Skip to main content
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www.anatomicaltravel.com/CB_site/Conception_to_birth3.htm Anatomical travel. Images of fertilisation and the development of the (human) embryo through to birth.
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www.uchsc.edu/ltc/fert.swf Fertilisation. A great animation of fertilisation, formation of the zygote and first mitotic division. A bit advanced but still worth watching.
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www.uclan.ac.uk/facs/health/n...anim/heart.swf Sonic. An animation showing the foetal blood circulation through the placenta to the changes allowing circulation through the lungs after birth.
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en.Wikipedia.org/wiki/Estrus Wikipedia. As always, good interesting information although some terms and concepts are beyond the requirements of this level.


7.11: Capacitance (Answers)
Check Your Understanding
8.1. \(\displaystyle 1.1×10^{−3}m\)
8.3. 3.59 cm, 17.98 cm
8.4. a. 25.0 pF;
b. 9.2
8.5. a. \(\displaystyle C=0.86pF,Q_1=10pC,Q_2=3.4pC,Q_3=6.8pC\);
b. \(\displaystyle C=2.3pF,Q_1=12pC,Q_2=Q_3=16pC\);
c. \(\displaystyle C=2.3pF,Q_1=9.0pC,Q_2=18pC,Q_3=12pC,Q_4=15pC\)
8.6. a.\(\displaystyle 4.0×10^{−13}J\); b. 9 times
8.7. a. 3.0; b. \(\displaystyle C=3.0C_0\)
8.9. a. \(\displaystyle C_0=20pF, C=42pF\);
b. \(\displaystyle Q_0=0.8nC, Q=1.7nC\);
c. \(\displaystyle V_0=V=40V\); d. \(\displaystyle U_0=16nJ, U=34nJ\)
Conceptual Questions
1. no; yes
3. false
5. no
7. \(\displaystyle 3.0μF,0.33μF\)
9. answers may vary
11. Dielectric strength is a critical value of an electrical field above which an insulator starts to conduct; a dielectric constant is the ratio of the electrical field in vacuum to the net electrical field in a material.
13. Water is a good solvent.
15. When energy of thermal motion is large (high temperature), an electrical field must be large too in order to keep electric dipoles aligned with it.
17. answers may vary
Problems
19. 21.6 mC
21. 1.55 V
23. 25.0 nF
25. \(\displaystyle 1.1×10^{−3}m^2\)
27. 500 µC
29. 1:16
31. a. 1.07 nC;
b. 267 V, 133 V
33. \(\displaystyle 0.29μF\)
34. 500 capacitors; connected in parallel
35. \(\displaystyle 3.08μF\) (series) and \(\displaystyle 13.0μ\) (parallel)
37. \(\displaystyle 11.4μF\)
39. 0.89 mC; 1.78 mC; 444 V
41. \(\displaystyle 7.5μJ\)
43. a. 405 J; b. 90.0 mC
45. 1.15 J
47. a. \(\displaystyle 4.43×10^{−9}F\);
b. 0.453 V;
c. \(\displaystyle 4.53×10^{−10}J\);
d. no
49. 0.7 mJ
51. a. 7.1 pF;
b. 42 pF
53. a. before 3.00 V; after 0.600 V;
b. before 1500 V/m; after 300 V/m
55. a. 3.91;
b. 22.8 V
57. a. 37 nC;
b. 0.4 MV/m;
c. 19 nC
59. a. \(\displaystyle 4.4μF\);
b. \(\displaystyle 4.0×10^{-5}C\)
61. \(\displaystyle 0.0135m^2\)
63. \(\displaystyle 0.185μJ\)
Additional Problems
65. a. 0.277 nF;
b. 27.7 nC;
c. 50 kV/m
67. a. 0.065 F;
b. 23,000 C;
c. 4.0 GJ
69. a. \(\displaystyle 75.6μC\); b. 10.8 V
71. a. 0.13 J;
b. no, because of resistive heating in connecting wires that is always present, but the circuit schematic does not indicate resistors
73. a. \(\displaystyle −3.00μF\);
b. You cannot have a negative \(\displaystyle C_2\) capacitance.
c. The assumption that they were hooked up in parallel, rather than in series, is incorrect. A parallel connection always produces a greater capacitance, while here a smaller capacitance was assumed. This could only happen if the capacitors are connected in series.
75. a. 14.2 kV;
b. The voltage is unreasonably large, more than 100 times the breakdown voltage of nylon.
c. The assumed charge is unreasonably large and cannot be stored in a capacitor of these dimensions.
Challenge Problems
77. a. 89.6 pF;
b. 6.09 kV/m;
c. 4.47 kV/m;
d. no
79. a. 421 J;
b. 53.9 mF
81. \(\displaystyle C=ε_0A/(d_1+d_2)\)
83. proof
Contributors and Attributions
Samuel J. Ling (Truman State University), Jeff Sanny (Loyola Marymount University), and Bill Moebs with many contributing authors. This work is licensed by OpenStax University Physics under a
Creative Commons Attribution License (by 4.0)
.


29.4A: Sprain and Strain
A strain is a tear in tendon fibers resulting from overstretching, while a sprain is an equivalent injury to a ligament.
- Describe a strain and its treatment
Key Points
- Strains and sprains can happen to anyone, but athletes are at higher risk.
- P.R.I.C.E. is a treatment method for muscular strain: protection, rest, ice, compression, elevation.
- Cold compression therapy uses ice packs and wrapping to reduce swelling and inflammation to encourage the healing process.
Key Terms
- strain : An injury to a muscle or tendon in which the muscle fibers tear as a result of overstretching.
- sprain : To weaken a ligament by sudden and excessive exertion, as by wrenching; to overstrain, or stretch injuriously, but without luxation; as, to sprain one’s ankle.
EXAMPLES
Gymnasts frequently suffer from mild ankle sprains. Cold compression therapy is the recommended treatment. As soon as possible, the gymnast needs to begin to take brief walks as part of rehabilitative therapy. Failure to rehabilitate the ankle can permanently weaken it.
A strain is an injury to a muscle or tendon in which the muscle fibers tear as a result of overstretching. A strain is also colloquially known as a pulled muscle. The equivalent injury to a ligament is a sprain. Typical symptoms of a strain include: localized pain, stiffness, discoloration, and bruising around the strained muscle. Strains can happen while doing everyday tasks and are not restricted to athletes. Nevertheless, people who play sports are more at risk of developing a strain due to increased muscle use.
Muscle Strain : Pulled hamstring after four days. Two images of the same leg. One of the pictures was taken in a mirror.
Treatment of Strains and Sprains
The first-line treatment for a muscular strain in the acute phase includes five steps commonly known as P.R.I.C.E.
- Protection: Apply soft padding to minimize impact with objects.
- Rest: Don’t use the injured limb.
- Ice: Apply ice to reduce swelling by reducing blood flow to the injury site. Never ice for more than 20 minutes at a time.
- Compression: Wrap the strained area to reduce swelling with an ACE soft-wrapped bandage.
- Elevation: Keep the strained area as close to the level of the heart as is conveniently possible to keep blood from pooling in the injured area.
The ice and compression (cold compression therapy) will stop the pain and swelling while the injury starts to heal itself. Controlling the inflammation is critical to the healing process and the icing further restricts fluid leaking into the injured area as well as controlling pain.
This immediate treatment is usually accompanied by the use of non-steroidal anti-inflammatory drugs (e.g., ibuprofen), which both reduce the immediate inflammation and relieve pain. However, NSAIDs, including aspirin and ibuprofen, affect platelet function (this is why they are known as “blood thinners”) and should not be taken during the period when tissue is bleeding because they will tend to increase blood flow and inhibit clotting. After the bleeding has stopped, NSAIDs can be used with some effectiveness to reduce inflammation and pain. It is recommended that the injured person should consult a medical provider if the injury is accompanied by severe pain, if the limb cannot be used, or if there is noticeable tenderness over an isolated spot. These can be signs of a broken or fractured bone, a sprain, or a complete muscle tear.


4.3: The Milgram Experiment
To demonstrate the ease with which power can be used to coerce people, Stanley Milgram conducted a scientific experiment that demonstrated how far people will go when confronted with someone who has power and is in a position of authority. In this instance, subjects often performed actions that were unethical when ordered to by a person in authority. Milgram’s experiment demonstrated the power of authority and how someone in a position of authority can influence people to behave unethically and against their wishes.
Milgram’s Studies on Obedience to Authority
The powerful ability of those in authority to control others was demonstrated in a remarkable set of studies performed by Stanley Milgram (1963). Milgram was interested in understanding the factors that lead people to obey the orders given by people in authority. He designed a study in which he could observe the extent to which a person who presented himself as an authority would be able to produce obedience, even to the extent of leading people to cause harm to others.
Like his professor Solomon Asch, Milgram’s interest in social influence stemmed in part from his desire to understand how the presence of a powerful person—particularly the German dictator Adolf Hitler who ordered the killing of millions of people during World War II—could produce obedience. Under Hitler’s direction, the German SS troops oversaw the execution of 6 million Jews as well as other “undesirables,” including political and religious dissidents, homosexuals, mentally and physically disabled people, and prisoners of war. Milgram used newspaper ads to recruit men (and in one study, women) from a wide variety of backgrounds to participate in his research. When the research participant arrived at the lab, he or she was introduced to a man who the participant believed was another research participant but who was actually an experimental confederate. The experimenter explained that the goal of the research was to study the effects of punishment on learning. After the participant and the confederate both consented to participate in the study, the researcher explained that one of them would be randomly assigned to be the teacher and the other the learner. They were each given a slip of paper and asked to open it and to indicate what it said. In fact both papers read teacher , which allowed the confederate to pretend that he had been assigned to be the learner and thus to assure that the actual participant was always the teacher. While the research participant (now the teacher) looked on, the learner was taken into the adjoining shock room and strapped to an electrode that was to deliver the punishment. The experimenter explained that the teacher’s job would be to sit in the control room and to read a list of word pairs to the learner. After the teacher read the list once, it would be the learner’s job to remember which words went together. For instance, if the word pair was blue-sofa , the teacher would say the word blue on the testing trials and the learner would have to indicate which of four possible words ( house , sofa , cat , or carpet ) was the correct answer by pressing one of four buttons in front of him. After the experimenter gave the “teacher” a sample shock (which was said to be at 45 volts) to demonstrate that the shocks really were painful, the experiment began. The research participant first read the list of words to the learner and then began testing him on his learning.
The shock panel, as shown in [the figure], “The Shock Apparatus Used in Milgram’s Obedience Study,” was presented in front of the teacher, and the learner was not visible in the shock room. The experimenter sat behind the teacher and explained to him that each time the learner made a mistake the teacher was to press one of the shock switches to administer the shock. They were to begin with the smallest possible shock (15 volts) but with each mistake the shock was increased by one level (an additional 15 volts).
Once the learner (who was, of course, actually an experimental confederate) was alone in the shock room, he unstrapped himself from the shock machine and brought out a tape recorder that he used to play a prerecorded series of responses that the teacher could hear through the wall of the room. As you can see in [the figure], “ The Confederate’s Schedule of Protest in the Milgram Experiments,” the teacher heard the learner say “ugh!” after the first few shocks. After the next few mistakes, when the shock level reached 150 volts, the learner was heard to exclaim “Get me out of here, please. My heart’s starting to bother me. I refuse to go on. Let me out!” As the shock reached about 270 volts, the learner’s protests became more vehement, and after 300 volts the learner proclaimed that he was not going to answer any more questions. From 330 volts and up the learner was silent. The experimenter responded to participants’ questions at this point, if they asked any, with a scripted response indicating that they should continue reading the questions and applying increasing shock when the learner did not respond.
| 75 volts | Ugh! |
| 90 volts | Ugh! |
| 105 volts | Ugh! ( louder ) |
| 120 volts | Ugh! Hey, this really hurts. |
| 135 volts | Ugh!! |
| 150 volts | Ugh!! Experimenter! That’s all. Get me out of here. I told you I had heart trouble. My heart’s starting to bother me now. Get me out of here, please. My heart’s starting to bother me. I refuse to go on. Let me out! |
| 165 volts | Ugh! Let me out! ( shouting ) |
| 180 volts | Ugh! I can’t stand the pain. Let me out of here! ( shouting ) |
| 195 volts | Ugh! Let me out of here! Let me out of here! My heart’s bothering me. Let me out of here! You have no right to keep me here! Let me out! Let me out of here! Let me out! Let me out of here! My heart’s bothering me. Let me out! Let me out! |
| 210 volts | Ugh!! Experimenter! Get me out of here. I’ve had enough. I won’t be in the experiment any more. |
| 225 volts | Ugh! |
| 240 volts | Ugh! |
| 255 volts | Ugh! Get me out of here. |
| 270 volts | ( agonized scream ) Let me out of here. Let me out of here. Let me out of here. Let me out. Do you hear? Let me out of here. |
| 285 volts | ( agonized scream ) |
| 300 volts | ( agonized scream ) I absolutely refuse to answer any more. Get me out of here. You can’t hold me here. Get me out. Get me out of here. |
| 315 volts | ( intensely agonized scream ) Let me out of here. Let me out of here. My heart’s bothering me. Let me out, I tell you. ( hysterically ) Let me out of here. Let me out of here. You have no right to hold me here. Let me out! Let me out! Let me out! Let me out of here! Let me out! Let me out! |
Before Milgram conducted his study, he described the procedure to three groups—college students, middle-class adults, and psychiatrists—asking each of them if they thought they would shock a participant who made sufficient errors at the highest end of the scale (450 volts). One hundred percent of all three groups thought they would not do so. He then asked them what percentage of “other people” would be likely to use the highest end of the shock scale, at which point the three groups demonstrated remarkable consistency by all producing (rather optimistic) estimates of around 1% to 2%.
The results of the actual experiments were themselves quite shocking. Although all of the participants gave the initial mild levels of shock, responses varied after that. Some refused to continue after about 150 volts, despite the insistence of the experimenter to continue to increase the shock level. Still others, however, continued to present the questions, and to administer the shocks, under the pressure of the experimenter, who demanded that they continue. In the end, 65% of the participants continued giving the shock to the learner all the way up to the 450 volts maximum, even though that shock was marked as “danger: severe shock,” and there had been no response heard from the participant for several trials. In sum, almost two-thirds of the men who participated had, as far as they knew, shocked another person to death, all as part of a supposed experiment on learning.
Milgram’s study is important in a law enforcement context for the following reasons:
- Officers must be careful in exercising authority, especially to those that are most vulnerable.
- Officers can also be greatly influenced by the negative/unethical actions of fellow officers and their own supervisors. It is important for senior officers to understand that Milgram’s study strongly suggests that the actions of senior officers will coerce the same action in junior officers. While senior officers may think they are not being copied, or are manipulating the junior officer, Milgram’s study suggests that they may be doing so.
- Law enforcement officers are commonly involved in extraordinary situations, where heightened stress and perceived danger are high. In this environment, even those most strong-willed individuals may be vulnerable to coercion.
- When a person is being arrested, his or her perception of losing freedom may provoke a reaction to the officer, despite the officer’s position of power.
It is important for any person who possesses power to understand and be aware of the coercive nature of power; that power and authority are easily used to make people do things they otherwise would not do. It is within this paradigm, that abuse of power can occur, and officers must be aware of their power and the ease with which it can be abused.
Text Attribution
The following description of Milgram’s experiment comes from the chapter “ Obedience, Power, and Leadership ” from the open textbook book Principles of Social Psychology: 1st International Edition. and is licensed CC BY 4.0 . All references cited on this page can be found at the end of the chapter of Obedience Power and Leadership .
Media Attributions
- Milgram_Experiment_v2 © Fred the Oyster is licensed under a CC BY-SA (Attribution ShareAlike) license


6.2: Pythagorean theorem
A triangle is called right if one of its angles is right. The side opposite the right angle is called the hypotenuse. The sides adjacent to the right angle are called legs.
Assume \(\triangle ABC\) is a right triangle with the right angle at \(C\). Then
\(AC^2 + BC^2 = AB^2.\)
- Proof
-
Let \(D\) be the foot point of \(C\) on \((AB)\).
According to Lemma 5.5.1 ,
\(AD < AC < AB\)
and
\(BD < BC < AB.\)
Therefore, \(D\) lies between \(A\) and \(B\); in particular,
\[AD + BD = AB.\]
Note that by the AA similarity condition, we have
\(\triangle ADC \sim \triangle ACB \sim \triangle CDB.\)
In particular,
\[\dfrac{AD}{AC} = \dfrac{AC}{AB} \text{ and } \dfrac{BD}{BC} = \dfrac{BC}{BA}.\]
Let us rewrite the two identities in 6.2.2:
\(AC^2 = AB \cdot AD\) and \(BC^2 = AB \cdot BD\).
Summing up these two identities and applying 6.2.1, we get that
\(AC^2 + BC^2 = AB \cdot (AD + BD) = AB^2.\)
Assume \(A, B, C\), and \(D\) are as in the proof above. Show that
\(CD^2 = AD \cdot BD.\)
The following exercise is the converse to the Pythagorean theorem.
- Hint
-
Apply that \(\triangle ADC \sim \triangle CDB\).
Assume that \(ABC\) is a triangle such that
\(AC^2 + BC^2 = AB^2.\)
Prove that the angle at \(C\) is right.
- Hint
-
Apply the Pythagorean theorem \(\PageIndex{1}\) and the SSS congruence condition


24.8: Nutrition and Diet
By the end of this section, you will be able to:
- Explain how different foods can affect metabolism
- Describe a healthy diet, as recommended by the U.S. Department of Agriculture (USDA)
- List reasons why vitamins and minerals are critical to a healthy diet
The carbohydrates, lipids, and proteins in the foods you eat are used for energy to power molecular, cellular, and organ system activities. Importantly, the energy is stored primarily as fats. The quantity and quality of food that is ingested, digested, and absorbed affects the amount of fat that is stored as excess calories.  Eating too much or too little food can lead to serious medical issues, including cardiovascular disease, cancer, anorexia, and diabetes, among others. Combine an unhealthy diet with unhealthy environmental conditions, such as smoking, and the potential medical complications increase significantly.
Food and Metabolism
The amount of energy that is needed or ingested per day is measured in calories. The nutritional Calorie (C) is the amount of heat it takes to raise 1 kg (1000 g) of water by 1 °C. This is different from the calorie (c) used in the physical sciences, which is the amount of heat it takes to raise 1 g of water by 1 °C. When we refer to "calorie," we are referring to the nutritional Calorie.
On average, a person needs 1500 to 2000 calories per day to sustain (or carry out) daily activities. The total number of calories needed by one person is dependent on their body mass, age, height, gender, activity level, and the amount of exercise per day. If exercise is regular part of one’s day, more calories are required. As a rule, people underestimate the number of calories ingested and overestimate the amount they burn through exercise. This can lead to ingestion of too many calories per day. The accumulation of an extra 3500 calories adds one pound of weight. If an excess of 200 calories per day is ingested, one extra pound of body weight will be gained every 18 days. At that rate, an extra 20 pounds can be gained over the course of a year. Of course, this increase in calories could be offset by increased exercise. Running or jogging one mile burns almost 100 calories.
The type of food ingested also affects the body’s metabolic rate. Processing of carbohydrates requires less energy than processing of proteins. In fact, the breakdown of carbohydrates requires the least amount of energy, whereas the processing of proteins demands the most energy. In general, the amount of calories ingested and the amount of calories burned determines the overall weight. To lose weight, the number of calories burned per day must exceed the number ingested. Calories are in almost everything you ingest, so when considering calorie intake, beverages must also be considered.
To help provide guidelines regarding the types and quantities of food that should be eaten every day, the USDA has updated their food guidelines from MyPyramid to MyPlate. They have put the recommended elements of a healthy meal into the context of a place setting of food. MyPlate categorizes food into the standard six food groups: fruits, vegetables, grains, protein foods, dairy, and oils. The accompanying website gives clear recommendations regarding quantity and type of each food that you should consume each day, as well as identifying which foods belong in each category. The accompanying graphic (Figure \(\PageIndex{1}\) ) gives a clear visual with general recommendations for a healthy and balanced meal. The guidelines recommend to “Make half your plate fruits and vegetables.” The other half is grains and protein, with a slightly higher quantity of grains than protein. Dairy products are represented by a drink, but the quantity can be applied to other dairy products as well.
ChooseMyPlate.gov provides extensive online resources for planning a healthy diet and lifestyle, including offering weight management tips and recommendations for physical activity. It also includes the SuperTracker, a web-based application to help you analyze your own diet and physical activity.
Everyday Connection
Metabolism and Obesity
Obesity in the United States is epidemic. The rate of obesity has been steadily rising since the 1980s. In the 1990s, most states reported that less than 10 percent of their populations was obese, and the state with the highest rate reported that only 15 percent of their population was considered obese. By 2010, the U.S. Centers for Disease Control and Prevention reported that nearly 36 percent of adults over 20 years old were obese and an additional 33 percent were overweight, leaving only about 30 percent of the population at a healthy weight. These studies find the highest levels of obesity are concentrated in the southern states. They also find the level of childhood obesity is rising.
Obesity is defined by the body mass index (BMI) , which is a measure of a person's weight in kilograms divided by height in meters squared. The normal, or healthy, BMI range is between 18 and 24.9 kg/m 2 . Overweight is defined as a BMI of 25 to 29.9 kg/m 2 , and obesity is considered to be a BMI greater than 30 kg/m 2 . Obesity can arise from a number of factors, including overeating, poor diet, sedentary lifestyle, limited sleep, genetic factors, and even diseases or drugs. Severe obesity (morbid obesity) or long-term obesity can result in serious medical conditions, including coronary heart disease; type 2 diabetes; endometrial, breast, or colon cancer; hypertension (high blood pressure); dyslipidemia (high cholesterol or elevated triglycerides); stroke; liver disease; gall bladder disease; sleep apnea or respiratory diseases; osteoarthritis; and infertility. Research has shown that losing weight can help reduce or reverse the complications associated with these conditions.
Vitamins
Vitamins are organic compounds found in foods and are a necessary part of the biochemical reactions in the body. They are involved in a number of processes, including mineral and bone metabolism, and cell and tissue growth, and they act as cofactors for energy metabolism. The B vitamins play the largest role of any vitamins in metabolism (Table \(\PageIndex{1}\) and Table \(\PageIndex{2}\)).
You get most of your vitamins through your diet, although some can be formed from the precursors absorbed during digestion. For example, the body synthesizes vitamin A from the β-carotene in orange vegetables like carrots and sweet potatoes. Vitamins are either fat-soluble or water-soluble. Fat-soluble vitamins A, D, E, and K, are absorbed through the intestinal tract with lipids in chylomicrons. Vitamin D is also synthesized in the skin through exposure to sunlight. Because they are carried in lipids, fat-soluble vitamins can accumulate in the lipids stored in the body. If excess vitamins are retained in the lipid stores in the body, hypervitaminosis can result.
Water-soluble vitamins, including the eight B vitamins and vitamin C, are absorbed with water in the gastrointestinal tract. These vitamins move easily through bodily fluids, which are water based, so they are not stored in the body. Excess water-soluble vitamins are excreted in the urine. Therefore, hypervitaminosis of water-soluble vitamins rarely occurs, except with an excess of vitamin supplements.
| Vitamin and alternative name | Sources | Recommended daily allowance | Function | Problems associated with deficiency |
|---|---|---|---|---|
|
A
retinal or β-carotene |
Yellow and orange fruits and vegetables, dark green leafy vegetables, eggs, milk, liver | 700–900 µ g | Eye and bone development, immune function | Night blindness, epithelial changes, immune system deficiency |
|
D
cholecalciferol |
Dairy products, egg yolks; also synthesized in the skin from exposure to sunlight | 5–15 µ g | Aids in calcium absorption, promoting bone growth | Rickets, bone pain, muscle weakness, increased risk of death from cardiovascular disease, cognitive impairment, asthma in children, cancer |
|
E
tocopherols |
Seeds, nuts, vegetable oils, avocados, wheat germ | 15 mg | Antioxidant | Anemia |
|
K
phylloquinone |
Dark green leafy vegetables, broccoli, Brussels sprouts, cabbage | 90–120 µ g | Blood clotting, bone health | Hemorrhagic disease of newborn in infants; uncommon in adults |
| Vitamin and alternative name | Sources | Recommended daily allowance | Function | Problems associated with deficiency |
|---|---|---|---|---|
|
B
1
thiamine |
Whole grains, enriched bread and cereals, milk, meat | 1.1–1.2 mg | Carbohydrate metabolism | Beriberi, Wernicke-Korsakoff syndrome |
|
B
2
riboflavin |
Brewer’s yeast, almonds, milk, organ meats, legumes, enriched breads and cereals, broccoli, asparagus | 1.1–1.3 mg | Synthesis of FAD for metabolism, production of red blood cells | Fatigue, slowed growth, digestive problems, light sensitivity, epithelial problems like cracks in the corners of the mouth |
|
B
3
niacin |
Meat, fish, poultry, enriched breads and cereals, peanuts | 14–16 mg | Synthesis of NAD, nerve function, cholesterol production | Cracked, scaly skin; dementia; diarrhea; also known as pellagra |
|
B
5
pantothenic acid |
Meat, poultry, potatoes, oats, enriched breads and cereals, tomatoes | 5 mg | Synthesis of coenzyme A in fatty acid metabolism | Rare: symptoms may include fatigue, insomnia, depression, irritability |
|
B
6
pyridoxine |
Potatoes, bananas, beans, seeds, nuts, meat, poultry, fish, eggs, dark green leafy vegetables, soy, organ meats | 1.3–1.5 mg | Sodium and potassium balance, red blood cell synthesis, protein metabolism | Confusion, irritability, depression, mouth and tongue sores |
|
B
7
biotin |
Liver, fruits, meats | 30 µ g | Cell growth, metabolism of fatty acids, production of blood cells | Rare in developed countries; symptoms include dermatitis, hair loss, loss of muscular coordination |
|
B
9
folic acid |
Liver, legumes, dark green leafy vegetables, enriched breads and cereals, citrus fruits | 400 µ g | DNA/protein synthesis | Poor growth, gingivitis, appetite loss, shortness of breath, gastrointestinal problems, mental deficits |
|
B
12
cyanocobalamin |
Fish, meat, poultry, dairy products, eggs | 2.4 µ g | Fatty acid oxidation, nerve cell function, red blood cell production | Pernicious anemia, leading to nerve cell damage |
|
C
ascorbic acid |
Citrus fruits, red berries, peppers, tomatoes, broccoli, dark green leafy vegetables | 75–90 mg | Necessary to produce collagen for formation of connective tissue and teeth, and for wound healing | Dry hair, gingivitis, bleeding gums, dry and scaly skin, slow wound healing, easy bruising, compromised immunity; can lead to scurvy |
Minerals
Minerals in food are inorganic compounds that work with other nutrients to ensure the body functions properly. Minerals cannot be made in the body; they come from the diet. The amount of minerals in the body is small—only 4 percent of the total body mass—and most of that consists of the minerals that the body requires in moderate quantities: potassium, sodium, calcium, phosphorus, magnesium, and chloride.
The most common minerals in the body are calcium and phosphorous, both of which are stored in the skeleton and necessary for the hardening of bones. Most minerals are ionized, and their ionic forms are used in physiological processes throughout the body. Sodium and chloride ions are electrolytes in the blood and extracellular tissues, and iron ions are critical to the formation of hemoglobin. There are additional trace minerals that are still important to the body’s functions, but their required quantities are much lower.
Like vitamins, minerals can be consumed in toxic quantities (although it is rare). A healthy diet includes most of the minerals your body requires, so supplements and processed foods can add potentially toxic levels of minerals. Table \(\PageIndex{3}\) and Table \(\PageIndex{4}\) provide a summary of minerals and their function in the body.
| Mineral | Sources | Recommended daily allowance | Function | Problems associated with deficiency |
|---|---|---|---|---|
| Potassium | Meats, some fish, fruits, vegetables, legumes, dairy products | 4700 mg | Nerve and muscle function; acts as an electrolyte | Hypokalemia: weakness, fatigue, muscle cramping, gastrointestinal problems, cardiac problems |
| Sodium | Table salt, milk, beets, celery, processed foods | 2300 mg | Blood pressure, blood volume, muscle and nerve function | Rare |
| Calcium | Dairy products, dark green leafy vegetables, blackstrap molasses, nuts, brewer’s yeast, some fish | 1000 mg | Bone structure and health; nerve and muscle functions, especially cardiac function | Slow growth, weak and brittle bones |
| Phosphorous | Meat, milk | 700 mg | Bone formation, metabolism, ATP production | Rare |
| Magnesium | Whole grains, nuts, leafy green vegetables | 310–420 mg | Enzyme activation, production of energy, regulation of other nutrients | Agitation, anxiety, sleep problems, nausea and vomiting, abnormal heart rhythms, low blood pressure, muscular problems |
| Chloride | Most foods, salt, vegetables, especially seaweed, tomatoes, lettuce, celery, olives | 2300 mg | Balance of body fluids, digestion | Loss of appetite, muscle cramps |
| Mineral | Sources | Recommended daily allowance | Function | Problems associated with deficiency |
|---|---|---|---|---|
| Iron | Meat, poultry, fish, shellfish, legumes, nuts, seeds, whole grains, dark leafy green vegetables | 8–18 mg | Transport of oxygen in blood, production of ATP | Anemia, weakness, fatigue |
| Zinc | Meat, fish, poultry, cheese, shellfish | 8–11 mg | Immunity, reproduction, growth, blood clotting, insulin and thyroid function | Loss of appetite, poor growth, weight loss, skin problems, hair loss, vision problems, lack of taste or smell |
| Copper | Seafood, organ meats, nuts, legumes, chocolate, enriched breads and cereals, some fruits and vegetables | 900 µ g | Red blood cell production, nerve and immune system function, collagen formation, acts as an antioxidant | Anemia, low body temperature, bone fractures, low white blood cell concentration, irregular heartbeat, thyroid problems |
| Iodine | Fish, shellfish, garlic, lima beans, sesame seeds, soybeans, dark leafy green vegetables | 150 µ g | Thyroid function | Hypothyroidism: fatigue, weight gain, dry skin, temperature sensitivity |
| Sulfur | Eggs, meat, poultry, fish, legumes | None | Component of amino acids | Protein deficiency |
| Fluoride | Fluoridated water | 3–4 mg | Maintenance of bone and tooth structure | Increased cavities, weak bones and teeth |
| Manganese | Nuts, seeds, whole grains, legumes | 1.8–2.3 mg | Formation of connective tissue and bones, blood clotting, sex hormone development, metabolism, brain and nerve function | Infertility, bone malformation, weakness, seizures |
| Cobalt | Fish, nuts, leafy green vegetables, whole grains | None | Component of B 12 | None |
| Selenium | Brewer’s yeast, wheat germ, liver, butter, fish, shellfish, whole grains | 55 µ g | Antioxidant, thyroid function, immune system function | Muscle pain |
| Chromium | Whole grains, lean meats, cheese, black pepper, thyme, brewer’s yeast | 25–35 µ g | Insulin function | High blood sugar, triglyceride, and cholesterol levels |
| Molybdenum | Legumes, whole grains, nuts | 45 µ g | Cofactor for enzymes | Rare |


7.11: Recent Research on Motivation Theories
Describe the modern advancements in the study of human motivation.
Employee motivation continues to be a major focus in organizational behavior. 35 We briefly summarize current motivation research here.
Content Theories
There is some interest in testing content theories (including Herzberg’s two-factor theory), especially in international research. Need theories are still generally supported, with most people identifying such workplace factors as recognition, advancement, and opportunities to learn as the chief motivators for them. This is consistent with need satisfaction theories. However, most of this research does not include actual measures of employee performance. Thus, questions remain about whether the factors that employees say motivate them to perform actually do.
Operant Conditioning Theory
There is considerable interest in operant conditioning theory, especially within the context of what has been called organizational behavior modification. Oddly enough, there has not been much research using operant conditioning theory in designing reward systems, even though there are obvious applications. Instead, much of the recent research on operant conditioning focuses on punishment and extinction. These studies seek to determine how to use punishment appropriately. Recent results still confirm that punishment should be used sparingly, should be used only after extinction does not work, and should not be excessive or destructive.
Equity Theory
Equity theory continues to receive strong research support. The major criticism of equity theory, that the inputs and outcomes people use to evaluate equity are ill-defined, still holds. Because each person defines inputs and outcomes, researchers are not in a position to know them all. Nevertheless, for the major inputs (performance) and outcomes (pay), the theory is a strong one. Major applications of equity theory in recent years incorporate and extend the theory into the area called organizational justice. When employees receive rewards (or punishments), they evaluate them in terms of their fairness (as discussed earlier). This is distributive justice. Employees also assess rewards in terms of how fair the processes used to distribute them are. This is procedural justice. Thus during organizational downsizing, when employees lose their jobs, people ask whether the loss of work is fair (distributive justice). But they also assess the fairness of the process used to decide who is laid off (procedural justice). For example, layoffs based on seniority may be perceived as more fair than layoffs based on supervisors’ opinions.
Goal Theory
It remains true that difficult, specific goals result in better performance than easy and vague goals, assuming they are accepted. Recent research highlights the positive effects of performance feedback and goal commitment in the goal-setting process. Monetary incentives enhance motivation when they are tied to goal achievement, by increasing the level of goal commitment. There are negative sides to goal theory as well. If goals conflict, employees may sacrifice performance on important job duties. For example, if both quantitative and qualitative goals are set for performance, employees may emphasize quantity because this goal achievement is more visible.
Expectancy Theory
The original formulation of expectancy theory specifies that the motivational force for choosing a level of effort is a function of the multiplication of expectancies and valences. Recent research demonstrates that the individual components predict performance just as well, without being multiplied. This does not diminish the value of expectancy theory. Recent research also suggests that high performance results not only when the valence is high, but also when employees set difficult goals for themselves.
One last comment on motivation: As the world of work changes, so will the methods organizations use to motivate employees. New rewards—time off instead of bonuses; stock options; on-site gyms, cleaners, and dental services; opportunities to telecommute; and others—will need to be created in order to motivate employees in the future. One useful path that modern researchers can undertake is to analyze the previous studies and aggregate the findings into more conclusive understanding of the topic through meta-analysis studies. 36
Catching the Entrepreneurial Spirit
Entrepreneurs and Motivation
Motivation can be difficult to elicit in employees. So what drives entrepreneurs, who by definition have to motivate themselves as well as others? While everyone from Greek philosophers to football coaches warns about undirected passion, a lack of passion will likely kill any start-up. An argument could be made that motivation is simply part of the discipline or the outcome of remaining fixed on a purpose to mentally remind yourself of why you get up in the morning.
Working from her home in Egypt, at age 30 Yasmine El-Mehairy launched Supermama.me, a start-up aimed at providing information to mothers throughout the Arab world. When the company began, El-Mehairy worked full time at her day job and 60 hours a week after that getting the site established. She left her full-time job to manage the site full time in January 2011, and the site went live that October. El-Mehairy is motivated to keep moving forward, saying that if she stops, she might not get going again (Knowledge @ Wharton 2012).
For El-Mehairy, the motivation didn’t come from a desire to work for a big company or travel the world and secure a master’s degree from abroad. She had already done that. Rather, she said she was motivated to “do something that is useful and I want to do something on my own” (Knowledge @ Wharton 2012 n.p.).
Lauren Lipcon, who founded a company called Injury Funds Now, attributes her ability to stay motivated to three factors: purpose, giving back, and having fun outside of work. Lipcon believes that most entrepreneurs are not motivated by money, but by a sense of purpose. Personally, she left a job with Arthur Andersen to begin her own firm out of a desire to help people. She also thinks it is important for people to give back to their communities because the change the entrepreneur sees in the community loops back, increasing motivation and making the business more successful. Lipcon believes that having a life outside of work helps keep the entrepreneur motivated. She particularly advocates for physical activity, which not only helps the body physically, but also helps keep the mind sharp and able to focus (Rashid 2017).
But do all entrepreneurs agree on what motivates them? A July 17, 2017 survey on the hearpreneur blog site asked 23 different entrepreneurs what motivated them. Seven of the 23 referred to some sense of purpose in what they were doing as a motivating factor, with one response stressing the importance of discovering one’s “personal why.” Of the remaining entrepreneurs, answers varied from keeping a positive attitude (three responses) and finding external sources (three responses) to meditation and prayer (two responses). One entrepreneur said his greatest motivator was fear: the fear of being in the same place financially one year in the future “causes me to take action and also alleviates my fear of risk” (Hear from Entrepreneurs 2017 n.p.). Only one of the 23 actually cited money and material success as a motivating factor to keep working.
However it is described, entrepreneurs seem to agree that passion and determination are key factors that carry them through the grind of the day-to-day.
Sources:
Hear from Entrepreneurs. 2017. “23 Entrepreneurs Explain Their Motivation or if ‘Motivation is Garbage.’” https://hear.ceoblognation.com/2017/...ation-garbage/
Knowledge @ Wharton. 2012. “The Super-motivated Entrepreneur Behind Egypt’s SuperMama.” http://knowledge.wharton.upenn.edu/a...pts-supermama/
Rashid, Brian. 2017. “How This Entrepreneur Sustains High Levels of Energy and Motivation.” Forbes. https://www.forbes.com/sites/brianra.../#2a8ec5591111
Questions:
- In the article from Hear from Entrepreneurs, one respondent called motivation “garbage”? Would you agree or disagree, and why?
- How is staying motivated as an entrepreneur similar to being motivated to pursue a college degree? Do you think the two are related? How?
- How would you expect motivation to vary across cultures?[/BOX]
Exercise \(\PageIndex{1}\)
- Understand the modern approaches to motivation theory.


1.2: Safety and Compliance in the Human Physiology Laboratory
This laboratory course requires that students fully participate in human physiology experiments or assessments each week. Because of this, several weeks require students to pay specific attention to dress code and other instructions that could affect the performance or maintenance of the equipment in the laboratory.
- Do not take apart or unplug any physiology equipment. Do not disconnect sensors from iWorx hardware boxes without assistance from your instructor.
- Do not disconnect any sensors from a subject’s body by tugging on the wires. Instead, carefully unsnap the sensors from the electrode patch.
- Adhesive from electrode patches is to be placed in the specified waste bins, only.
- Phone use in lab is prohibited except when a specific application is requested for an experiment (Slack, phone timer, etc.).
- Do not use lab computers for accessing unauthorized web content (any web content not required for the class).
- Before lab, do not apply lotion or makeup on bodily areas where electrodes or other equipment will be placed.
- Please follow weekly dress code requirements provided in the course announcements.
SAFETY REGULATIONS FOR THE PHYSIOLOGY LABORATORY
- Note the locations of lab safety equipment: emergency shower and eyewash station, fire extinguishers, and first aid kit.
- Do not eat, drink, handle contact lenses, store food or beverages, or apply cosmetics in the laboratory unless you are instructed to. Please keep personal water bottles inside your bag.
- Students who are pregnant or who have any other medical conditions (e.g. pacemaker or other implanted device) that might necessitate special precautions during lab exercises must inform the instructor.
- Use safety glasses in experiments involving the use of hazardous liquids.
- Wear disposable gloves when handling blood or contacting body fluids.
- Keep all liquids away from the edge of the table to avoid spills. Keep all liquids away from electronic hardware and outlets. Report any spills or accidents to your instructor immediately.
- Dispose of waste properly. Sharps (syringes, scalpel blades, etc.), glass, materials contaminated with body fluids, and animal tissues have specific disposal bins in the A&P laboratories. Do not put these in the trash.
- Dispose of electrode backing and electrodes in the small can at each station. Fold over used electrodes so the sticky side is adhered to itself and place in small cans. Do not put large objects in the small cans; dispose of these in the regular trash.
- Clean up and organize at the end of every lab period. Put away physiology lab equipment at your station so that it is in the same condition you found it.
COVID-19 Precautions
During COVID-19 protocols the laboratory has specific additional guidelines to deter the spread of virus in shared university spaces. From Iowa State EH&S: Physical distancing (six feet/two meters) during the period of community transmission of COVID-19 can be challenging for laboratories, given that researchers might not have dedicated workspaces and may have to use shared equipment.
- Assume that any person you come into contact with could be an asymptomatic carrier of COVID-19.
- Per the recommendations of the CDC and government mandates, maintaining six feet of physical distancing is important to reduce the risks of transmitting COVID-19.
- Appropriate face coverings should be worn at all times in the laboratory setting. Face coverings must be worn appropriately, with the mask covering both the nose and mouth.
- Wash hands with soap and water before and after each laboratory session.
- Follow the guidelines for entering and exiting the laboratory in order to avoid close contact with others.
- Follow all safety guidelines as specified by your instructor each week in the laboratory. If you do not follow safety guidelines, the instructor may dismiss you from class.
- DO NOT come to the laboratory if you feel ill. You may join class virtually or complete an alternative activity. You may not schedule an in-person make-up lab due to contact tracing and room capacity guidelines.
- Students should help clean and disinfect shared laboratory equipment with the provided materials at each workstation, including disinfecting computer keyboard covers. DO NOT spray any cleaning/disinfecting product directly onto electronics.
- Dispose of single use cleaning materials in the regular garbage can, NOT in the small cans for electrodes at each station.
For additional information pertaining to laboratory safety at ISU, please see the following publication: Laboratory Safety Manual [pdf]
Link here for general safety/emergency response information: Student Emergency and Safety Information (external link)


10.4B: Compromised Host
- Discuss the risk factors that contribute to the acquiring of nosocomial infections or hospital-acquired infections
Susceptible Hosts
A nosocomial infection, also known as a hospital-acquired infection or HAI, is an infection whose development is favoured by a hospital environment, such as one acquired by a patient during a hospital visit or one developing among hospital staff. Such infections include fungal and bacterial infections. They are aggravated by the reduced resistance of individual patients. Numerous risk factors in the hospital setting predispose a patient to infection. These risk factors can broadly be divided into three areas.
- People in hospitals are usually already in a ‘poor state of health’, impairing their defense against bacteria. Advanced age or premature birth, along with immunodeficiency (due to drugs, illness, or irradiation) present a general risk, while other diseases can present specific risks; for instance, chronic obstructive pulmonary disease can increase chances of respiratory tract infection.
- Invasive devices, for instance intubation tubes, catheters, surgical drains, and tracheostomy tubes all bypass the body’s natural lines of defense against pathogens and provide an easy route for infection. Patients already colonized at the time of admission are instantly put at greater risk when they undergo invasive procedures.
- Patients’ treatments can leave them vulnerable to infection: immunosuppression and antacid treatment undermine the body’s defences, while antimicrobial therapy (removing competitive flora and only leaving resistant organisms) and recurrent blood transfusions have also been identified as risk factors.
Prevention
Hospitals have sanitation protocols regarding uniforms, equipment sterilization, washing, and other preventive measures. Thorough hand washing and/or use of alcohol rubs by all medical personnel before and after each patient contact is one of the most effective ways to combat nosocomial infections. More careful use of antimicrobial agents, such as antibiotics, is also considered vital. Despite sanitation protocol, patients cannot be entirely isolated from infectious agents. Furthermore, patients are often prescribed antibiotics and other antimicrobial drugs to help treat illness; this can increase the selection pressure for the emergence of resistant strains.
Key Points
- People in hospitals are usually already in a “poor state of health,” impairing their defense against bacteria.
- Invasive devices bypass the body’s natural lines of defense against pathogens and provide an easy route for infection.
- Patients’ treatments can leave them vulnerable to infection.
Key Terms
- infection : An uncontrolled growth of harmful microorganisms in a host.
- defense : The action of defending or protecting from attack, danger, or injury.
- Invasive : Invasive species, also called invasive exotics or simply exotics, is a nomenclature term and categorization phrase used for flora and fauna, and for specific restoration-preservation processes in native habitats, with several definitions.


20: Carboxylic Acids and Nitriles
This chapter presents a straightforward discussion of the chemistry of carboxylic acids (formerly called “fatty acids”) and nitriles. As usual, we begin with a description of how the compounds are named. We then consider the subtleties of their structure, and how these structural features influence physical properties, such as boiling point. We place considerable emphasis on the dissociation of carboxylic acids and the effect of substituents on acid strength.
We have already encountered a number of methods of preparing carboxylic acids. We shall review these methods, and learn about two additional procedures. The only reactions of carboxylic acids that we study in detail in this chapter are reduction and decarboxylation, although two other common reactions of carboxylic acids, alpha substitution and nucleophilic acyl substitution, will be described in later chapters.
We will then look at the formation of nitriles and their chemical reactivity; and our discussion of carboxylic acid and nitrile chemistry concludes with a look at the infrared and NMR spectra of these compounds, with emphasis on the characterization of carboxylic acids.
-
- 20.5: Substituent Effects on Acidity
- You have already seen how the presence of an electron‑withdrawing or electron‑releasing group affects the stability of a positively charged carbocation. Now you see how these groups affect the stability of carboxylate anions, and in turn, determine the dissociation constant of a carboxylic acid.


10.4: Evaluate and Graph Logarithmic Functions
By the end of this section, you will be able to:
- Convert between exponential and logarithmic form
- Evaluate logarithmic functions
- Graph Logarithmic functions
- Solve logarithmic equations
- Use logarithmic models in applications
Before you get started, take this readiness quiz.
-
Solve: \(x^{2}=81\).
If you missed this problem, review Example 6.46. -
Evaluate: \(3^{−2}\).
If you missed this problem, review Example 5.15. -
Solve: \(2^{4}=3x−5\).
If you missed this problem, review Example 2.2.
We have spent some time finding the inverse of many functions. It works well to ‘undo’ an operation with another operation. Subtracting ‘undoes’ addition, multiplication ‘undoes’ division, taking the square root ‘undoes’ squaring.
As we studied the exponential function, we saw that it is one-to-one as its graphs pass the horizontal line test. This means an exponential function does have an inverse. If we try our algebraic method for finding an inverse, we run into a problem.
\(f(x)=a^{x}\)
Rewrite with \(y=f(x)\).
\(y=a^{x}\)
Interchange the variables \(x\) and \(y\).
\(x=a^{y}\)
Solve for \(y\).
Oops! We have no way to solve for \(y\)!
To deal with this we define the logarithm function with base a to be the inverse of the exponential function \(f(x)=a^{x}\). We use the notation \(f^{−1}(x)=log_{a}x\) and say the inverse function of the exponential function is the logarithmic function.
The function \(f(x)=\log_{a}x\) is the logarithmic function with base \(a\), where \(a>0,x>0\), and \(a≠1\).
\(y=\log _{a} x\) is equivalent to \(x=a^{y}\)
Convert Between Exponential and Logarithmic Form
Since the equations \(y=\log _{a} x\) and \(x=a^{y}\) are equivalent, we can go back and forth between them. This will often be the method to solve some exponential and logarithmic equations. To help with converting back and forth let’s take a close look at the equations. See Figure 10.3.1 . Notice the positions of the exponent and base.
If we realize the logarithm is the exponent it makes the conversion easier. You may want to repeat, “base to the exponent give us the number.”
Convert to logarithmic form:
- \(2^{3}=8\)
- \(5^{\frac{1}{2}}=\sqrt{5}\)
- \(\left(\frac{1}{2}\right)^{x}=\frac{1}{16}\)
Solution :
Convert to logarithmic form:
- \(3^{2}=9\)
- \(7^{\frac{1}{2}}=\sqrt{7}\)
- \(\left(\frac{1}{3}\right)^{x}=\frac{1}{27}\)
- Answer
-
- \(\log _{3} 9=2\)
- \(\log _{7} \sqrt{7}=\frac{1}{2}\)
- \(\log _{\frac{1}{3}} \frac{1}{27}=x\)
Convert to logarithmic form:
- \(4^{3}=64\)
- \(4^{\frac{1}{3}}=\sqrt[3]{4}\)
- \(\left(\frac{1}{2}\right)^{x}=\frac{1}{32}\)
- Answer
-
- \(\log _{4} 64=3\)
- \(\log _{4} \sqrt[3]{4}=\frac{1}{3}\)
- \(\log _{\frac{1}{2}} \frac{1}{32}=x\)
In the next example we do the reverse—convert logarithmic form to exponential form.
Convert to exponential form:
- \(2=\log _{8} 64\)
- \(0=\log _{4} 1\)
- \(-3=\log _{10} \frac{1}{1000}\)
Solution :
Convert to exponential form:
- \(3=\log _{4} 64\)
- \(0=\log _{x} 1\)
- \(-2=\log _{10} \frac{1}{100}\)
- Answer
-
- \(64=4^{3}\)
- \(1=x^{0}\)
- \(\frac{1}{100}=10^{-2}\)
Convert to exponential form:
- \(3=\log _{3} 27\)
- \(0=\log _{x} 1\)
- \(-1=\log _{10} \frac{1}{10}\)
- Answer
-
- \(27=3^{3}\)
- \(1=x^{0}\)
- \(\frac{1}{10}=10^{-1}\)
Evaluate Logarithmic Functions
We can solve and evaluate logarithmic equations by using the technique of converting the equation to its equivalent exponential equation.
Find the value of \(x\):
- \(\log _{x} 36=2\)
- \(\log _{4} x=3\)
- \(\log _{\frac{1}{2}} \frac{1}{8}=x\)
Solution :
a.
\(\log _{x} 36=2\)
Convert to exponential form.
\(x^{2}=36\)
Solve the quadratic.
\(x=6, \quad \cancel{x=-6}\)
The base of a logarithmic function must be positive, so we eliminate \(x=−6\).
\(x=6 \quad\) Therefore, \(\log _{6} 36=2\)
b.
\(\log _{4} x=3\)
Convert to exponential form.
\(4^{3}=x\)
Simplify.
\(x=64 \quad\) Therefore \(, \log _{4} 64=3\)
c.
\(\log _{\frac{1}{2}} \frac{1}{8}=x\)
Convert to exponential form.
\(\left(\frac{1}{2}\right)^{x}=\frac{1}{8}\)
Rewrite \(\frac{1}{8}\) as \(\left(\frac{1}{2}\right)^{3}\).
\(\left(\frac{1}{2}\right)^{x}=\left(\frac{1}{2}\right)^{3}\)
With the same base, the exponents must be equal.
\(x=3 \quad\) Therefore \(\log _{\frac{1}{2}} \frac{1}{8}=3\)
Find the value of \(x\):
- \(\log _{x} 64=2\)
- \(\log _{5} x=3\)
- \(\log _{\frac{1}{2}} \frac{1}{4}=x\)
- Answer
-
- \(x=8\)
- \(x=125\)
- \(x=2\)
Find the value of \(x\):
- \(\log _{x} 81=2\)
- \(\log _{3} x=5\)
- \(\log _{\frac{1}{3}} \frac{1}{27}=x\)
- Answer
-
- \(x=9\)
- \(x=243\)
- \(x=3\)
When see an expression such as \(log_{3}27\), we can find its exact value two ways. By inspection we realize it means “\(3\) to what power will be \(27\)”? Since \(3^{3}=27\), we know \(log_{3}27=3\). An alternate way is to set the expression equal to \(x\) and then convert it into an exponential equation.
Find the exact value of each logarithm without using a calculator:
- \(\log _{5} 25\)
- \(\log _{9} 3\)
- \(\log _{2} \frac{1}{16}\)
Solution :
a.
\(\log _{5} 25\)
\(5\) to what power will be \(25\)?
\(\log _{5} 25=2\)
Or
Set the expression equal to \(x\).
\(\log _{5} 25=x\)
Change to exponential form.
\(5^{x}=25\)
Rewrite \(25\) as \(5^{2}\).
\(5^{x}=5^{2}\)
With the same base the exponents must be equal.
\(x=2 \quad\) Therefore \(, \log _{5} 25=2\).
b.
\(\log _{9} 3\)
Set the expression equal to \(x\).
\(\log _{9} 3=x\)
Change to exponential form.
\(9^{x}=3\)
Rewrite \(9\) as \(3^{2}\).
\(\left(3^{2}\right)^{x}=3^{1}\)
Simplify the exponents.
\(3^{2 x}=3^{1}\)
With the same base the exponents must be equal.
\(2 x=1\)
Solve the equation.
\(x=\frac{1}{2} \quad\) Therefore \(, \log _{9} 3=\frac{1}{2}\).
c.
\(\log _{2} \frac{1}{16}\)
Set the expression equal to \(x\).
\(\log _{2} \frac{1}{16}=x\)
Change to exponential form.
\(2^{x}=\frac{1}{16}\)
Rewrite \(16\) as \(2^{4}\).
\(2^{x}=\frac{1}{2^{4}}\)
\(2^{x}=2^{-4}\)
With the same base the exponents must be equal.
\(x=-4 \quad\) Therefore \(, \log _{2} \frac{1}{16}=-4\).
Find the exact value of each logarithm without using a calculator:
- \(\log _{12} 144\)
- \(\log _{4} 2\)
- \(\log _{2} \frac{1}{32}\)
- Answer
-
- \(2\)
- \(\frac{1}{2}\)
- \(-5\)
Find the exact value of each logarithm without using a calculator:
- \(\log _{9} 81\)
- \(\log _{8} 2\)
- \(\log _{3} \frac{1}{9}\)
- Answer
-
- \(2\)
- \(\frac{1}{3}\)
- \(-2\)
Graph Logarithmic Functions
To graph a logarithmic function \(y=log_{a}x\), it is easiest to convert the equation to its exponential form, \(x=a^{y}\). Generally, when we look for ordered pairs for the graph of a function, we usually choose an \(x\)-value and then determine its corresponding \(y\)-value. In this case you may find it easier to choose \(y\)-values and then determine its corresponding \(x\)-value.
Graph \(y=\log _{2} x\).
Solution :
To graph the function, we will first rewrite the logarithmic equation, \(y=\log _{2} x\), in exponential form, \(2^{y}=x\).
We will use point plotting to graph the function. It will be easier to start with values of \(y\) and then get \(x\).
| \(y\) | \(2^{y}=x\) | \((x,y)\) |
|---|---|---|
| \(-2\) | \(2^{-2}=\frac{1}{2^{2}}=\frac{1}{4}\) | \((\frac{1}{4},2)\) |
| \(-1\) | \(2^{-1}=\frac{1}{2^{1}}=\frac{1}{2}\) | \((\frac{1}{2},-1)\) |
| \(0\) | \(2^{0}=1\) | \((1,0)\) |
| \(1\) | \(2^{1}=2\) | \((2,1)\) |
| \(2\) | \(2^{2}=4\) | \((4,2)\) |
| \(3\) | \(2^{3}=8\) | \((8,3)\) |
Graph: \(y=\log _{3} x\).
- Answer
Graph: \(y=\log _{5} x\).
- Answer
The graphs of \(y=\log _{2} x, y=\log _{3} x\), and \(y=\log _{5} x\) are the shape we expect from a logarithmic function where \(a>1\).
We notice that for each function the graph contains the point \((1,0)\). This make sense because \(0=log_{a}1\) means \(a^{0}=1\) which is true for any \(a\).
The graph of each function, also contains the point \((a,1)\). This makes sense as \(1=\log _{a} a\) means \(a^{1}=a\). which is true for any \(a\).
Notice too, the graph of each function \(y=\log _{a} x\) also contains the point \(\left(\frac{1}{a},-1\right)\). This makes sense as \(-1=\log _{a} \frac{1}{a}\) means \(a^{-1}=\frac{1}{a}\), which is true for any \(a\).
Look at each graph again. Now we will see that many characteristics of the logarithm function are simply ’mirror images’ of the characteristics of the corresponding exponential function.
What is the domain of the function? The graph never hits the \(y\)-axis. The domain is all positive numbers. We write the domain in interval notation as \((0,∞)\).
What is the range for each function? From the graphs we can see that the range is the set of all real numbers. There is no restriction on the range. We write the range in interval notation as \((−∞,∞)\).
When the graph approaches the \(y\)-axis so very closely but will never cross it, we call the line \(x=0\), the \(y\)-axis, a vertical asymptote.
| Domain | \((0, \infty)\) |
| Range | \((-\infty, \infty)\) |
| \(x\)-intercept | \((1,0)\) |
| \(y\)-intercept | None |
| Contains | \((a, 1),\left(\frac{1}{a},-1\right)\) |
| Asymptote | \(y\)-axis |
Our next example looks at the graph of \(y=log_{a}x\) when \(0<a<1\).
Graph \(y=\log _{\frac{1}{3}} x\).
Solution :
To graph the function, we will first rewrite the logarithmic equation, \(y=\log _{\frac{1}{3}} x\), in exponential form, \(\left(\frac{1}{3}\right)^{y}=x\).
We will use point plotting to graph the function. It will be easier to start with values of \(y\) and then get \(x\).
| \(y\) | \(\left(\frac{1}{3}\right)^{y}=x\) | \((x,y)\) |
|---|---|---|
| \(-2\) | \(\left(\frac{1}{3}\right)^{-2}=3^{2}=9\) | \((9,-2)\) |
| \(-1\) | \(\left(\frac{1}{3}\right)^{-1}=3^{1}=3\) | \((3,-1)\) |
| \(0\) | \(\left(\frac{1}{3}\right)^{0}=1\) | \((1,0)\) |
| \(1\) | \(\left(\frac{1}{3}\right)^{1}=\frac{1}{3}\) | \(\left(\frac{1}{3}, 1\right)\) |
| \(2\) | \(\left(\frac{1}{3}\right)^{2}=\frac{1}{9}\) | \(\left(\frac{1}{9}, 2\right)\) |
| \(3\) | \(\left(\frac{1}{3}\right)^{3}=\frac{1}{27}\) | \(\left(\frac{1}{27}, 3\right)\) |
Graph: \(y=\log _{\frac{1}{2}} x\).
- Answer
Graph: \(y=\log _{\frac{1}{4}} x\).
- Answer
Now, let’s look at the graphs \(y=\log _{\frac{1}{2}} x, y=\log _{\frac{1}{3}} x\) and \(y=\log _{\frac{1}{4}} x\), so we can identify some of the properties of logarithmic functions where \(0<a<1\).
The graphs of all have the same basic shape. While this is the shape we expect from a logarithmic function where \(0<a<1\).
We notice, that for each function again, the graph contains the points,\((1,0),(a, 1),\left(\frac{1}{a},-1\right)\). This make sense for the same reasons we argued above.
We notice the domain and range are also the same—the domain is \((0,∞)\) and the range is \((−∞,∞)\). The \(y\)-axis is again the vertical asymptote.
We will summarize these properties in the chart below. Which also include when \(a>1\).
| When \(a>1\) | When \(0<a<1\) | ||
|---|---|---|---|
| 1\)">Domain | \((0, \infty)\) | Domain | \((0, \infty)\) |
| 1\)">Range | \((-\infty, \infty)\) | Range | \((-\infty, \infty)\) |
| 1\)">\(x\)-intercept | \((1,0)\) | \(x\)-intercept | \((1,0)\) |
| 1\)">\(y\)-intercept | None | \(y\)-intercept | None |
| 1\)">Contains | \((a, 1),\left(\frac{1}{a},-1\right)\) | Contains | \((a, 1),\left(\frac{1}{a},-1\right)\) |
| 1\)">Asymptote | \(y\) -axis | Asymptote | \(y\) -axis |
| 1\)">Basic shape | Increasing | Basic shape | Decreasing |
We talked earlier about how the logarithmic function \(f^{-1}(x)=\log _{a} x\) is the inverse of the exponential function \(f(x)=a^{x}\). The graphs in Figure 10.3.12 show both the exponential (blue) and logarithmic (red) functions on the same graph for both \(a>1\) and \(0<a<1\).
Notice how the graphs are reflections of each other through the line \(y=x\). We know this is true of inverse functions. Keeping a visual in your mind of these graphs will help you remember the domain and range of each function. Notice the \(x\)-axis is the horizontal asymptote for the exponential functions and the \(y\)-axis is the vertical asymptote for the logarithmic functions.
Solve Logarithmic Equations
When we talked about exponential functions, we introduced the number \(e\). Just as \(e\) was a base for an exponential function, it can be used a base for logarithmic functions too. The logarithmic function with base \(e\) is called the natural logarithmic function . The function \(f(x)=\log _{e} x\) is generally written \(f(x)=\ln x\) and we read it as “el en of \(x\).”
The function \(f(x)=\ln x\) is the natural logarithmic function with base \(e\), where \(x>0\).
\(y=\ln x\) is equivalent to \(x=e^{y}\)
When the base of the logarithm function is \(10\), we call it the common logarithmic function and the base is not shown. If the base \(a\) of a logarithm is not shown, we assume it is \(10\).
The function \(f(x)=\log x\) is the common logarithmic function with base \(10\), where \(x>0\).
\(y=\log x\) is equivalent to \(x=10^{y}\)
To solve logarithmic equations, one strategy is to change the equation to exponential form and then solve the exponential equation as we did before. As we solve logarithmic equations, \(y=log_{a}x\), we need to remember that for the base \(a\), \(a>0\) and \(a≠1\). Also, the domain is \(x>0\). Just as with radical equations, we must check our solutions to eliminate any extraneous solutions.
Solve:
- \(\log _{a} 49=2\)
- \(\ln x=3\)
Solution :
a.
\(\log _{a} 49=2\)
Rewrite in exponential form.
\(a^{2}=49\)
Solve the equation using the square root property.
\(a=\pm 7\)
The base cannot be negative, so we eliminate \(a=-7\).
\(a=7, \quad \cancel{a=-7}\)
Check. \(a=7\)
\(\begin{aligned} \log _{a} 49&=2 \\ \log_{7}49&\stackrel{?}{=}2 \\ 7^{2}&\stackrel{?}{=}49 \\ 49&=49 \end{aligned}\)
b.
\(\ln x=3\)
Rewrite in exponential form.
\(e^{3}=x\)
Check. \(x=e^{3}\)
\(\begin{aligned} \ln x &=3 \\ \ln e^{3} & \stackrel{?}{=} 3 \\ e^{3} &=e^{3} \end{aligned}\)
Solve:
- \(\log _{a} 121=2\)
- \(\ln x=7\)
- Answer
-
- \(a=11\)
- \(x=e^{7}\)
Solve:
- \(\log _{a} 64=3\)
- \(\ln x=9\)
- Answer
-
- \(a=4\)
- \(x=e^{9}\)
Solve:
- \(\log _{2}(3 x-5)=4\)
- \(\ln e^{2 x}=4\)
Solution :
a.
\(\log _{2}(3 x-5)=4\)
Rewrite in exponential form.
\(2^{4}=3 x-5\)
Simplify.
\(16=3 x-5\)
Solve the equation.
\(21=3 x\)
\(7=x\)
Check. \(x=7\)
\(\begin{aligned} \log _{2}(3 x-5)&=4 \\ \log_{2}(3\cdot7-5)&\stackrel{?}{=}4\\ \log_{2}(16)&\stackrel{?}{=}4 \\ 2^{4}& \stackrel{?}{=}16 \\ 16&=16 \end{aligned}\)
b.
\(\ln e^{2 x}=4\)
Rewrite in exponential form.
\(e^{4}=e^{2 x}\)
Since the bases are the same the exponents are equal.
\(4=2 x\)
Solve the equation.
\(2=x\)
Check. \(x=2\)
\(\begin{aligned} \ln e^{2 x} &=4 \\ \ln e^{2 \cdot 2} & \stackrel{?}{=} 4 \\ \ln e^{4} &=4 \\ e^{4} &=e^{4} \end{aligned}\)
Solve:
- \(\log _{2}(5 x-1)=6\)
- \(\ln e^{3 x}=6\)
- Answer
-
- \(x=13\)
- \(x=2\)
Solve:
- \(\log _{3}(4 x+3)=3\)
- \(\ln e^{4 x}=4\)
- Answer
-
- \(x=6\)
- \(x=1\)
Use Logarithmic Models in Applications
There are many applications that are modeled by logarithmic equations. We will first look at the logarithmic equation that gives the decibel (dB) level of sound. Decibels range from \(0\), which is barely audible to \(160\), which can rupture an eardrum. The \(10^{−12}\) in the formula represents the intensity of sound that is barely audible.
Decibel Level of Sound
The loudness level, \(D\), measured in decibels, of a sound of intensity, \(I\), measured in watts per square inch is
\(D=10 \log \left(\frac{I}{10^{-12}}\right)\)
Extended exposure to noise that measures \(85\) dB can cause permanent damage to the inner ear which will result in hearing loss. What is the decibel level of music coming through ear phones with intensity \(10^{−2}\) watts per square inch?
Solution :
| Substitute in the intensity level, \(I\). | |
| Simplify. | |
| Since \(\log 10^{10}=10\). | |
| Multiply. | |
| The decibel level of music coming through earphones is \(100\) dB. |
What is the decibel level of one of the new quiet dishwashers with intensity \(10^{−7}\) watts per square inch?
- Answer
-
The quiet dishwashers have a decibel level of \(50\) dB.
What is the decibel level heavy city traffic with intensity \(10^{−3}\) watts per square inch?
- Answer
-
The decibel level of heavy traffic is \(90\) dB.
The magnitude \(R\) of an earthquake is measured by a logarithmic scale called the Richter scale. The model is \(R=\log I\), where \(I\) is the intensity of the shock wave. This model provides a way to measure earthquake intensity .
The magnitude \(R\) of an earthquake is measured by\(R=\log I\), where \(I\) is the intensity of its shock wave.
In 1906, San Francisco experienced an intense earthquake with a magnitude of \(7.8\) on the Richter scale. Over \(80\)% of the city was destroyed by the resulting fires. In 2014, Los Angeles experienced a moderate earthquake that measured \(5.1\) on the Richter scale and caused $\(108\) million dollars of damage. Compare the intensities of the two earthquakes.
Solution :
To compare the intensities, we first need to convert the magnitudes to intensities using the log formula. Then we will set up a ratio to compare the intensities.
Convert the magnitudes to intensities.
\(R=\log I\)
1906 earthquake
\(7.8=\log I\)
Convert to exponential form.
\(I=10^{7.8}\)
2014 earthquake
\(5.1=\log I\)
Convert to exponential form.
\(I=10^{5.1}\)
Form a ratio of the intensities.
\(\frac{\text { Intensity for } 1906}{\text { Intensity for } 2014}\)
Substitute in the values.
\(\frac{10^{7.8}}{10^{5.1}}\)
Divide by subtracting the exponents.
\(10^{2.7}\)
Evaluate.
\(501\)
The intensity of the 1906 earthquake was about \(501\) times the intensity of the 2014 earthquake.
In 1906, San Francisco experienced an intense earthquake with a magnitude of \(7.8\) on the Richter scale. In 1989, the Loma Prieta earthquake also affected the San Francisco area, and measured \(6.9\) on the Richter scale. Compare the intensities of the two earthquakes.
- Answer
-
The intensity of the 1906 earthquake was about \(8\) times the intensity of the 1989 earthquake.
In 2014, Chile experienced an intense earthquake with a magnitude of \(8.2\) on the Richter scale. In 2014, Los Angeles also experienced an earthquake which measured \(5.1\) on the Richter scale. Compare the intensities of the two earthquakes.
- Answer
-
The intensity of the earthquake in Chile was about \(1,259\) times the intensity of the earthquake in Los Angeles.
Access these online resources for additional instruction and practice with evaluating and graphing logarithmic functions.
Key Concepts
- Properties of the Graph of \(y=\log _{a} x\):
| When \(a>1\) | When \(0<a<1\) | ||
|---|---|---|---|
| 1\)">Domain | \((0, \infty)\) | Domain | \((0, \infty)\) |
| 1\)">Range | \((-\infty, \infty)\) | Range | \((-\infty, \infty)\) |
| 1\)">\(x\)-intercept | \((1,0)\) | \(x\)-intercept | \((1,0)\) |
| 1\)">\(y\)-intercept | None | \(y\)-intercept | None |
| 1\)">Contains | \((a, 1),\left(\frac{1}{a},-1\right)\) | Contains | \((a, 1),\left(\frac{1}{a},-1\right)\) |
| 1\)">Asymptote | \(y\) -axis | Asymptote | \(y\) -axis |
| 1\)">Basic shape | Increasing | Basic shape | Decreasing |
- Decibel Level of Sound: The loudness level, \(D\), measured in decibels, of a sound of intensity, \(I\), measured in watts per square inch is \(D=10 \log \left(\frac{I}{10^{-12}}\right)\).
- Earthquake Intensity: The magnitude \(R\) of an earthquake is measured by \(R=\log I\), where \(I\) is the intensity of its shock wave.
Glossary
- common logarithmic function
-
The function \(f(x)=\log x\) is the common logarithmic function with base \(10\), where \(x>0\).
\(y=\log x\) is equivalent to \(x=10^{y}\)
- logarithmic function
-
The function \(f(x)=\log _{a} x\) is the logarithmic function with base \(a\), where \(a>0,x>0\), and \(a≠1\).
\(y=\log _{a} x\) is equivalent to \(x=a^{y}\)
- natural logarithmic function
-
The function \(f(x)=\ln x\) is the natural logarithmic function with base \(e\), where \(x>0\).
\(y=\ln x\) is equivalent to \(x=e^{y}\)


17.4: Graphing Relations
Recall that if \(R\) is a relation on a set \(A\text{,}\) then formally \(R\) is a subset \(A \times A\text{.}\) In other words, \(R\) is a collection of ordered pairs of elements from \(A\text{.}\)
Also recall that in a directed graph, the edge collection is formally defined to be a collection of ordered pairs of vertices. So when the set \(A\) is finite, we may regard \(A\) as a set of vertices and \(R\) as a collection of (directed) edges in a graph!
To summarize, we may represent a relation \(R \subseteq A \times A\) by the directed graph \((A,R)\text{.}\) The vertices of the graph are the elements of \(A\text{,}\) and for elements \(a_1,a_2 \in A\text{,}\) we draw an arrow from \(a_1\) to \(a_2\) if \(a_1 \mathrel{R} a_2\) is true.
Recall that for natural numbers \(m\) and \(n\text{,}\) \(m \mid n\) means “\(m\) divides \(n\)”. Consider this relation on the finite set \(A = \{ 2,3,4,5,6,7,8,9,10 \}\text{.}\) The graph of this relation appears in Figure \(\PageIndex{1}\).
Note that each vertex has a loop since every number divides itself.
Using the same division relation on the same set \(A\) as in Example \(\PageIndex{1}\) above, we may obtain the graph for the inverse relation by just reversing the direction of all the arrows in the graph in Figure \(\PageIndex{1}\).
How are the properties of a relation reflected in its graph?
Reflexive relations.
In this case, \(a \mathrel{R} a\) is true for every element \(a \in A\text{,}\) so every vertex has a loop. For example, the relation in Example \(\PageIndex{1}\) is reflexive, and we see this mirrored in the graph in Figure \(\PageIndex{1}\) by the placement of a loop at every node.
When a relation is understood to be reflexive, we often omit the loops from its graph to reduce visual clutter.
Symmetric relations.
In this case, the conditional \(a_1 \mathrel{R} a_2 \Rightarrow a_2 \mathrel{R} a_1\) is always true. Therefore, in the graph for \(R\text{,}\) whenever we have an arrow from \(a_1\) to \(a_2\text{,}\) we must also have an arrow from \(a_2\) to \(a_1\text{.}\)
On the set \(A = \{a,b,c,d\}\text{,}\) the relation
\begin{equation*} R = \{(a,b),(b,a),(b,c),(c,b)\} \end{equation*}
is symmetric, and we see this property reflected in the graph in Figure \(\PageIndex{2}\), as each pair of related (distinct) nodes has an arrow in each direction between them.
When \(R\) is symmetric, arrows are essentially meaningless since between every pair of vertices we will have either no arrows or one arrow in each direction. So we may as well draw the graph for \(R\) as an ordinary (undirected) graph instead of a directed graph, replacing each pair of arrows with a single edge.
The relation in the previous example is more concisely depicted graphically as in Figure \(\PageIndex{3}\) below.
Antisymmetric relations.
In this case, we never have both \(a_1 \mathrel{R} a_2\) and \(a_2 \mathrel{R} a_1\) for \(a_1 \ne a_2\text{,}\) so in the graph for \(R\text{,}\) no pair of vertices can have two oppositely-directed arrows between them.
On the set \(A = \{a,b,c,d\}\text{,}\) the relation
\begin{equation*} R = \{(a,a),(a,b),(c,b)\} \end{equation*}
is antisymmetric, and we see this property reflected in the graph in Figure \(\PageIndex{4}\), as each pair of distinct nodes has at most one arrow between them.
Transitive relations.
In this case, the conditional \(a_1 \mathrel{R} a_2 \land a_2 \mathrel{R} a_3 \Rightarrow a_1 \mathrel{R} a_3\) is always true. Therefore, in the graph for \(R\text{,}\) every “chain” of two arrows has a corresponding “composite” arrow.
On the set \(A = \{a,b,c,d,e\} \text{,}\) the relation
\begin{equation*} R = \{(a,b),(b,c),(a,c),(d,e),(e,d),(d,d),(e,e)\} \end{equation*}
is transitive, and we see this property reflected in the graph in Figure \(\PageIndex{5}\), as each pair of arrows forming a “chain” between three nodes has a corresponding “composite” arrow from the first node in the chain to the third.
In the graph of a transitive relation, we often omit the “composite” arrows to reduce visual clutter, as we can infer from “chains” of arrows where the “composite” arrows would go. For example, we did this in both the power set graph in Example 14.4.1 (see Figure 14.4.1 ) and in the division graph in Example 14.4.2 (see Figure 14.4.2 ). It should be obvious that the relations “is a subset of” and “divides” are transitive, so there was no need to clutter up the graphs of those relations with extra “composite” arrows — we could trace the fact that one set was a subset of another or the fact that one number divides another by following a chain of arrows through intermediate nodes, as necessary.


5.1: Introduction to the Workforce Planning Process
What you’ll learn to do: Describe the steps in the workforce planning process
Workforce planning is a process for aligning human resources—as a function and the people—in support of the organization’s strategy over a long-term planning horizon. In this section, we’ll discuss what the workforce planning process entails, including its alignment with business strategy, how human resources are assessed, how to determine the supply and demand for labor and the outcome of the workforce planning process.
Contributors and Attributions
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