8.6: Logistic Growth
In our basic exponential growth scenario, we had a recursive equation of the form
\(P_{n}=P_{n-1}+r P_{n-1}\)
In a confined environment, however, the growth rate may not remain constant. In a lake, for example, there is some maximum sustainable population of fish, also called a carrying capacity .
The carrying capacity , or maximum sustainable population , is the largest population that an environment can support.
For our fish, the carrying capacity is the largest population that the resources in the lake can sustain. If the population in the lake is far below the carrying capacity, then we would expect the population to grow essentially exponentially. However, as the population approaches the carrying capacity, there will be a scarcity of food and space available, and the growth rate will decrease. If the population exceeds the carrying capacity, there won’t be enough resources to sustain all the fish and there will be a negative growth rate, causing the population to decrease back to the carrying capacity.
If the carrying capacity was 5000, the growth rate might vary something like that in the graph shown. Note that this is a linear equation with intercept at \(0.1\) and slope \(-\frac{0.1}{5000}\), so we could write an equation for this adjusted growth rate as:
\(r_{\text {adjusted}}=0.1-\frac{0.1}{5000} P=0.1\left(1-\frac{P}{5000}\right)\)
Substituting this in to our original exponential growth model for \(r\) gives
\(P_{n}=P_{n-1}+0.1\left(1-\frac{P_{n-1}}{5000}\right) P_{n-1}\)
If a population is growing in a constrained environment with carrying capacity \(K\), and absent constraint would grow exponentially with growth rate \(r\), then the population behavior can be described by the logistic growth model:
\(P_{n}=P_{n-1}+r\left(1-\frac{P_{n-1}}{K}\right) P_{n-1}\)
Unlike linear and exponential growth, logistic growth behaves differently if the populations grow steadily throughout the year or if they have one breeding time per year. The recursive formula provided above models generational growth, where there is one breeding time per year (or, at least a finite number); there is no explicit formula for this type of logistic growth.
A forest is currently home to a population of 200 rabbits. The forest is estimated to be able to sustain a population of 2000 rabbits. Absent any restrictions, the rabbits would grow by 50% per year. Predict the future population using the logistic growth model.
Solution
Modeling this with a logistic growth model, \(r = 0.50\), \(K = 2000\), and \(P_0 = 200\). Calculating the next year:
\(P_{1}=P_{0}+0.50\left(1-\frac{P_{0}}{2000}\right) P_{0}=200+0.50\left(1-\frac{200}{2000}\right) 200=290\)
We can use this to calculate the following year:
\(P_{2}=P_{1}+0.50\left(1-\frac{P_{1}}{2000}\right) P_{1}=290+0.50\left(1-\frac{290}{2000}\right) 290 \approx 414\)
A calculator was used to compute several more values:
\(\begin{array}{|l|l|l|l|l|l|l|l|l|l|l|l|}
\hline n & 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 \\
\hline P_{n} & 200 & 290 & 414 & 578 & 784 & 1022 & 1272 & 1503 & 1690 & 1821 & 1902 \\
\hline
\end{array}\)
Plotting these values, we can see that the population starts to increase faster and the graph curves upwards during the first few years, like exponential growth, but then the growth slows down as the population approaches the carrying capacity.
On an island that can support a population of 1000 lizards, there is currently a population of 600. These lizards have a lot of offspring and not a lot of natural predators, so have very high growth rate, around 150%. Calculating out the next couple generations:
Solution
\(P_{1}=P_{0}+1.50\left(1-\frac{P_{0}}{1000}\right) P_{0}=600+1.50\left(1-\frac{600}{1000}\right) 600=960\)
\(P_{2}=P_{1}+1.50\left(1-\frac{P_{1}}{1000}\right) P_{1}=960+1.50\left(1-\frac{960}{1000}\right) 960=1018\)
Interestingly, even though the factor that limits the growth rate slowed the growth a lot, the population still overshot the carrying capacity. We would expect the population to decline the next year.
\(P_{3}=P_{2}+1.50\left(1-\frac{P_{3}}{1000}\right) P_{3}=1018+1.50\left(1-\frac{1018}{1000}\right) 101\)
Calculating out a few more years and plotting the results, we see the population wavers above and below the carrying capacity, but eventually settles down, leaving a steady population near the carrying capacity.
A field currently contains 20 mint plants. Absent constraints, the number of plants would increase by 70% each year, but the field can only support a maximum population of 300 plants. Use the logistic model to predict the population in the next three years.
- Answer
-
\(P_{1}=P_{0}+0.70\left(1-\frac{P_{0}}{300}\right) P_{0}=20+0.70\left(1-\frac{20}{300}\right) 20=33\)
\(P_{2}=54\)
\(P_{3}=85\)
On a neighboring island to the one from the previous example, there is another population of lizards, but the growth rate is even higher – about 205%.
Solution
Calculating out several generations and plotting the results, we get a surprise: the population seems to be oscillating between two values, a pattern called a 2-cycle.
While it would be tempting to treat this only as a strange side effect of mathematics, this has actually been observed in nature. Researchers from the University of California observed a stable 2-cycle in a lizard population in California[1].
Taking this even further, we get more and more extreme behaviors as the growth rate increases higher. It is possible to get stable 4-cycles, 8-cycles, and higher. Quickly, though, the behavior approaches chaos (remember the movie Jurassic Park?).
[1] users.rcn.com/jkimball.ma.ult...ulations2.html


5.1.5: Team Diversity
-
- Last updated
- Save as PDF
Learning Objectives
- How does team diversity enhance decision-making and problem-solving?
Decision-making and problem-solving can be much more dynamic and successful when performed in a diverse team environment. The multiple diverse perspectives can enhance both the understanding of the problem and the quality of the solution. As I reflect on some of the leadership development work that I have done in my career, I can say from experience that the team activities and projects that intentionally brought diverse individuals together created the best environments for problem-solving. Diverse leaders from a variety of functions, from across the globe, at varying stages of their careers and experiences with and outside of the company had the most robust discussions and perspectives. Diversity is a word that is very commonly used today, but the importance of diversity and building diverse teams can sometimes get lost in the normal processes of doing business. Let’s discuss why we need to keep these principles front of mind.
In the Harvard Business Review article “Why Diverse Teams are Smarter” (Nov. 2016), David Rock and Heidi Grant support the idea that increasing workplace diversity is a good business decision. 9 A 2015 McKinsey report on 366 public companies found that those in the top quartile for ethnic and racial diversity in management were 35% more likely to have financial returns above their industry mean, and those in the top quartile for gender diversity were 15% more likely to have returns above the industry mean. Similarly, in a global analysis conducted by Credit Suisse, organizations with at least one female board member yielded a higher return on equity and higher net income growth than those that did not have any women on the board.
Additional research on diversity has shown that diverse teams are better at decision-making and problem-solving because they tend to focus more on facts, per the Rock and Grant article. 10 A study published in the Journal of Personality and Social Psychology showed that people from diverse backgrounds “might actually alter the behavior of a group’s social majority in ways that lead to improved and more accurate group thinking.” It turned out that in the study, the diverse panels raised more facts related to the case than homogenous panels and made fewer factual errors while discussing available evidence. Another study noted in the article showed that diverse teams are “more likely to constantly reexamine facts and remain objective. They may also encourage greater scrutiny of each member’s actions, keeping their joint cognitive resources sharp and vigilant. By breaking up workforce homogeneity, you can allow your employees to become more aware of their own potential biases—entrenched ways of thinking that can otherwise blind them to key information and even lead them to make errors in decision-making processes.” In other words, when people are among homogeneous and like-minded (nondiverse) teammates, the team is susceptible to groupthink and may be reticent to think about opposing viewpoints since all team members are in alignment. In a more diverse team with a variety of backgrounds and experiences, the opposing viewpoints are more likely to come out and the team members feel obligated to research and address the questions that have been raised. Again, this enables a richer discussion and a more in-depth fact-finding and exploration of opposing ideas and viewpoints in order to solve problems.
Diversity in teams also leads to greater innovation. A Boston Consulting Group article entitled “The Mix that Matters: Innovation through Diversity” explains a study in which BCG and the Technical University of Munich conducted an empirical analysis to understand the relationship between diversity in managers (all management levels) and innovation. The key findings of this study show that: 11
- The positive relationship between management diversity and innovation is statistically significant—and thus companies with higher levels of diversity derive more revenue from new products and services.
- The innovation boost isn’t limited to a single type of diversity. The presence of managers who are either female or are from other countries, industries, or companies can cause an increase in innovation.
- Management diversity seems to have a particularly positive effect on innovation at complex companies—those that have multiple product lines or that operate in multiple industry segments.
- To reach its potential, gender diversity needs to go beyond tokenism. In the study, innovation performance only increased significantly when the workforce included more than 20% women in management positions. Having a high percentage of female employees doesn’t increase innovation if only a small number of women are managers.
- At companies with diverse management teams, openness to contributions from lower-level workers and an environment in which employees feel free to speak their minds are crucial for fostering innovation.
When you consider the impact that diverse teams have on decision-making and problem-solving—through the discussion and incorporation of new perspectives, ideas, and data—it is no wonder that the BCG study shows greater innovation. Team leaders need to reflect upon these findings during the early stages of team selection so that they can reap the benefits of having diverse voices and backgrounds.
concept check
- Why do diverse teams focus more on data than homogeneous teams?
- How are diversity and innovation related?


18.10: Introduction to Global Employee Engagement
What you’ll learn to do: Describe ways to engage a global team
As we discussed in a previous module, employee engagement is extremely helpful in boosting company productivity and sales. However, what Americans perceive to be enticing and engaging may not be received the same way in other cultures. Global teams encompass a wide variety of cultural differences, and in order to engage employees throughout an entire international company, it is important to understand what motivates and excites them. Other key components like effective communication and a sense of trust are invaluable in building a highly engaged team. Teams that feel valued and trusted are more likely to perform at a higher level. Understanding how to bridge cultural differences and engage all team members is an important skill for global HR managers to possess.
Contributors and Attributions
- Introduction to Globalu00a0Employee Engagement. Authored by : Freedom Learning Group. Provided by : Lumen Learning. License : CC BY: Attribution
- Authored by : Free-Photos. Provided by : Pixa. Located at : Pixabay. License : CC0: No Rights Reserved . License Terms : Pixabay License


17.2: Class I- Sodium Channel Blockers
By the end of this section, you should be able to:
- 17.2.1 Identify the characteristics of the sodium channel blocker drugs used to treat dysrhythmias.
- 17.2.2 Explain the indications, actions, adverse reactions, and interactions of the sodium channel blocker drugs used to treat dysrhythmias.
- 17.2.3 Describe the nursing implications of the sodium channel blocker drugs used to treat dysrhythmias.
- 17.2.4 Explain the client education related to the sodium channel blocker drugs used to treat dysrhythmias.
Class I antidysrhythmic drugs are known as sodium channel blockers. They block sodium channels in conduction tissue and myocytes. Sodium channels are responsible for the depolarization of cardiac cells, and blocking these channels leads to slower conduction and a longer refractory period between impulses. Class I antidysrhythmic drugs are further broken down into classes IA, IB, and IC.
Class IA drugs cause moderate blockade of sodium channels. They also have an additional mechanism of action and block potassium channels. Blockage of potassium channels leads to a property called proarrhythmia , meaning the drugs have a propensity to cause other arrhythmias. Drugs that block potassium channels prolong the QT interval, which increases the risk for torsade de pointes.
Class IB drugs block sodium channels with mild intensity. These drugs are exclusively used for ventricular arrhythmias.
Class IC drugs strongly block sodium channels but do not block potassium channels; thus, they are less proarrhythmic than class IA agents (King et al., 2023).
Various sodium channel antidysrhythmic drugs exist. This chapter will cover the following drugs in more detail: quinidine, procainamide, lidocaine, mexiletine, flecainide, and propafenone.
Quinidine
Quinidine is a class IA antidysrhythmic drug. It is available as an oral immediate-release tablet and an extended-release tablet. These tablets are different salt forms of quinidine; the immediate-release form is quinidine sulfate, and the extended-release form is quinidine gluconate. It is important to note that these are not interchangeable.
Quinidine is approved by the U.S. Food and Drug Administration (FDA) for cardioversion of atrial fibrillation and flutter to sinus rhythm and then for maintenance of sinus rhythm after cardioversion. It is also approved for suppression of recurrent ventricular arrhythmias.
Quinidine can easily be confused with quinine; nurses must be careful to avoid mixing up these drugs. Clients should be educated on the potential for diarrhea and contact their provider if this occurs because it can lead to electrolyte abnormalities that can exacerbate arrhythmias. Clients who take quinidine should avoid grapefruit and grapefruit juice because of potential interactions.
Procainamide
Procainamide is a class IA antidysrhythmic drug that is available only in an intravenous form. It is FDA approved for life-threatening ventricular arrhythmias but is used off-label for other arrhythmias as well, including atrial fibrillation. Procainamide is metabolized to N-acetyl procainamide, which is also active.
Lidocaine and Mexiletine
Lidocaine is a class IB antiarrhythmic drug. It has various uses as an anesthetic; however, in terms of dysrhythmias, it is used exclusively for treating ventricular arrhythmias. It can be used both for acute treatment of a primary arrhythmia and in advanced cardiac life support when clients have sudden cardiac arrest due to ventricular arrhythmias. When used for treating arrhythmias, lidocaine is administered only intravenously.
Clinical Tip
Lidocaine
Because lidocaine may cause central nervous system toxicity, the nurse should assess the client frequently. Signs and symptoms to watch for include sedation and irritability (twitching), which could progress to convulsions and respiratory depression/arrest.
Safety Alert
Lidocaine
Lidocaine is available in many formulations, some of which are not suitable for intravenous administration. Lidocaine with epinephrine is one such dosage form. The Institute for Safe Medication Practices reported a death that occurred when a client was administered topical lidocaine with epinephrine instead of the intravenous form of lidocaine with epinephrine, causing fatal cardiac arrhythmias. Lidocaine with epinephrine should never be used as an antidysrhythmic agent.
Mexiletine is like lidocaine in that it is a class IB drug. It is FDA approved for managing ventricular arrhythmias and is available as an oral capsule for long-term use.
Flecainide and Propafenone
Flecainide is a class IC antidysrhythmic drug that is available as an oral tablet. It is used for clients with paroxysmal symptomatic supraventricular tachycardias, including atrial fibrillation and atrial flutter. It can also be used for life-threatening ventricular tachyarrhythmias.
Propafenone is also a class IC drug. It is available as an oral immediate-release tablet and an oral extended-release capsule. It is FDA approved to treat life-threatening ventricular arrhythmias and to prolong time to recurrence of paroxysmal atrial fibrillation/flutter in clients without structural heart disease. Propafenone is used off-label in a technique informally called “pill in a pocket.” This technique refers to a scenario in which a client who has symptomatic recurrent paroxysmal atrial fibrillation can keep propafenone on hand and take it as needed when they detect symptoms of atrial fibrillation. The medication will convert them back to sinus rhythm without their needing to visit a health care provider. They do not take the medication daily, just when they need it.
Table 17.2 lists common class I antidysrhythmic drugs and typical routes and dosing for adult clients.
| Drug | Routes and Dosage Ranges |
|---|---|
| Quinidine |
Ventricular arrhythmias:
Quinidine sulfate: 200–600 mg every 6–12 hours. Quinidine gluconate: 324–648 mg every 8–12 hours. |
| Procainamide |
Ventricular arrhythmias:
Loading dose: 100 mg intravenously (IV) every 5 minutes, administered until the arrhythmia is suppressed or 500 mg has been administered. Maintenance dose: Typically 50 mcg/min/kg. |
|
Lidocaine
( Xylocaine ) |
Ventricular arrhythmias (hemodynamically stable): 1–1.5 mg/kg IV bolus, repeat 0.5–0.75 mg/kg IV bolus every 5–10 minutes; maintenance infusion is 1–4 mg/minute IV. |
| Mexiletine | Ventricular arrhythmias: 150–200 mg orally every 8–12 hours. |
| Flecainide | Maintenance of normal sinus rhythm in atrial fibrillation: 50–200 mg orally every 12 hours. |
|
Propafenone
( Rythmol SR ) |
Maintenance of normal sinus rhythm in atrial fibrillation:
Immediate release: 150–300 mg orally every 8 hours. Extended release: 225–425 mg orally every 12 hours. |
Adverse Effects and Contraindications
All sodium channel blockers have some propensity to cause heart blocks; however, the adverse effect profile of sodium channel blockers varies greatly depending on the drug.
Quinidine is associated with many serious risks. Trials have shown increased mortality risk compared with placebo and other antiarrhythmic drugs, and for this reason the drug has a boxed warning (DailyMed, Quinidine gluconate, 2021). Quinidine has a high incidence of diarrhea as an adverse effect (greater than 20%). It is hepatotoxic and has been associated with serious liver problems, including granulomatous hepatitis, so it should be used with caution in clients with hepatic impairment. It can be associated with thrombocytopenia and therefore should not be used in clients with baseline thrombocytopenia. In addition to acting as a sodium channel blocker, it blocks potassium channels and prolongs the QT interval, which leads to a proarrhythmic effect. The nurse will need to monitor the client’s ECG, complete blood count (CBC), and liver and kidney function.
Many serious risks associated with the use of procainamide have led to several boxed warnings. It has the potential to cause drug–induced lupus erythematosus–like syndrome as well as blood dyscrasias such as potentially fatal agranulocytosis, and it has been associated with increased mortality compared with placebo in certain populations. In addition to acting as a sodium channel blocker, it blocks potassium channels and prolongs the QT interval, which leads to a proarrhythmic effect. It should not be used in clients with certain heart blocks or in those with systemic lupus erythematosus or torsade de pointes. Procainamide can cause hypotension, so the nurse must be mindful of maximum infusion rates to minimize this possibility. The nurse will also need to monitor the client’s ECG, CBC, and drug levels of procainamide and its active metabolite, N-acetyl procainamide.
Lidocaine can cause central nervous system toxicity. The nurse will need to monitor the client’s ECG continuously and may need to monitor the client’s blood levels for the drug, depending on how long it will be administered.
Flecainide carries a warning because it can cause 1:1 AV conduction in atrial fibrillation/flutter. This means that the ventricles will beat once for each impulse in the atria, which is so fast that it can cause hemodynamic collapse. The nurse should monitor the client’s ECG and any blood levels ordered.
Contraindications vary based on the drug. Some of the most noteworthy contraindications include:
- Quinidine and procainamide should not be used in clients with baseline prolonged QT interval, given their effect on the QT interval. They also should not be used with other drugs that strongly prolong the QT interval. Procainamide is contraindicated in clients who have systemic lupus erythematosus.
- Lidocaine is contraindicated in individuals with an arrhythmia called Wolff–Parkinson–White syndrome and in those with certain heart blocks.
- Flecainide and propafenone are contraindicated in clients with structural heart disease, such as heart failure or myocardial infarction, due to potentially fatal proarrhythmic effects and worsening of heart failure.
Table 17.3 is a drug prototype table for sodium channel blockers featuring the class IB drug lidocaine. It lists drug class, mechanism of action, adult dosage, indications, therapeutic effects, drug and food interactions, adverse effects, and contraindications.
|
Drug Class
Sodium channel blocker Mechanism of Action Blocks sodium channels in the cells of the cardiac conduction system |
Drug Dosage
Ventricular arrhythmias (hemodynamically stable): 1–1.5 mg/kg IV bolus, repeat 0.5–0.75 mg/kg IV bolus every 5–10 minutes; maintenance infusion is 1–4 mg/minute IV. |
|
Indications
Acute treatment of ventricular cardiac arrhythmias Local and regional anesthesia Therapeutic Effects Facilitates cardioversion Maintains normal sinus rhythm |
Drug Interactions
Digoxin (toxicity) Beta-adrenergic blockers Food Interactions No significant interactions |
|
Adverse Effects
Bradycardia Hemodynamic collapse Atrioventricular block Sinus arrest Delirium Psychosis Seizure Nausea Tinnitus Dyspnea Bronchospasm |
Contraindications
Hypersensitivity Severe sinoatrial node disfunction, aortic stenosis, or intraventricular blocks |
Nursing Implications
The nurse should do the following for clients who are taking sodium channel blockers:
- Recognize and monitor for serious and potentially dangerous adverse effects of these drugs.
- Monitor the ECG of any client who takes antiarrhythmic drugs, paying careful attention to the QT interval.
- Monitor blood levels of drugs and relevant metabolites.
- Provide client teaching regarding the drug and when to call the health care provider. See below for client teaching guidelines.
Client Teaching Guidelines
The client taking quinidine should:
- Report excessive diarrhea to their health care provider.
The client taking mexiletine should:
- Report signs and symptoms of hepatic toxicity, such as jaundice.
The client taking propafenone should:
- Report signs of infection due to agranulocytosis, such as fever or sore throat.
FDA Black Box Warning
Quinidine
Active antiarrhythmic therapy has resulted in increased mortality; the risk of active therapy is greatest in clients with structural heart disease.
Procainamide
Prolonged administration of procainamide often leads to the development of a positive antinuclear antibody test, with or without symptoms of lupus erythematosus–like syndrome. If a positive antinuclear antibody titer develops, the benefits versus the risks of continued procainamide therapy should be assessed. In addition, agranulocytosis, bone marrow depression, neutropenia, hypoplastic anemia, and thrombocytopenia in clients receiving procainamide HCl have been reported at a rate of approximately 0.5%.
Mexiletine
In postmarketing experience, abnormal liver function tests have been reported, some in the first few weeks of therapy.
Flecainide
A review of the world literature revealed reports of ventricular tachycardia in 0.4% of clients receiving flecainide. Of 19 clients reported in the literature with chronic atrial fibrillation, 10.5% experienced ventricular tachycardia or ventricular fibrillation. Flecainide is not recommended for use in clients with chronic atrial fibrillation.
Flecainide, Propafenone, Mexiletine, Procainamide
In the CAST trial, there was excessive mortality in post–myocardial infarction clients who were administered flecainide or another similar drug, encainide.


6.1: Frequency
Learning Objectives
- Explain time and clock
This chapter and the preceding one have good, solid physical titles. Inertia. Waves. But underlying the physical content is a thread of mathematics designed to teach you a language for describing spacetime. Without this language, the complications of relativity rapidly build up and become unmanageable. In section 5.2, we saw that there are physically compelling reasons for switching back and forth between different coordinate systems — different ways of attaching names to the events that make up spacetime. A toddler in a bilingual family gets a payoff for switching back and forth between asking Mamá in Spanish for dulces and alerting Daddy in English that Barbie needs to be rescued from falling off the couch. She may bounce back and forth between the two languages in a single sentence — a habit that linguists call “ code switching .” In relativity, we need to build ﬂuency in a language that lets us talk about actual phenomena without getting hung up on the naming system.
Is time’s ﬂow constant?
The simplest naming task is in \(0 + 1\) dimensions: a time-line like the ones in history class. If we name the points in time \(A, B, C, ...\) or \(1, 2, 3, ...\), or Bush, Clinton, Bush, ..., how do we know that we’re marking off equal time intervals? Does it make sense to imagine that time itself might speed up and slow down, or even start and stop? The second law of thermodynamics encourages us to think that it could. If the universe had existed for an infinite time, then entropy would have maximized itself — a long time ago, presumably — and we would not exist, because the heat death of the universe would already have happened.
Clock-comparison experiments
But what would it actually mean empirically for time’s rate of ﬂow to vary? Unless we can tie this to the results of experiments, it’s nothing but cut-rate metaphysics. In a Hollywood movie where time could stop, the scriptwriters would show us the stopping through the eyes of an observer, who would stroll past frozen waterfalls and snapshotted bullets in mid-ﬂight. The observer’s brain is a kind of clock, and so is the waterfall. We’re left with what’s known as a clock-comparison experiment. To date, all clock-comparison experiments have given null results. Matsakis et al . 1 found that pulsars match the rates of atomic clocks with a drift of less than about \(10^{-6}\) seconds over \(10\) years. Guéna et al . 2 observed that atomic clocks using atoms of different isotopes drifted relative to one another by no more than about \(10^{-16}\) per year. Any non-null result would have caused serious problems for relativity. One of the expectations in an Aristotelian description of spacetime is that the motion of material objects on earth would naturally slow down relative to celestial phenomena such as the rising and setting of the sun. The relativistic interpretation of time dilation as an effect on time itself also depends crucially on the null results of these experiments.
Birdtracks notation
As a simple example of clock comparison, let’s imagine using the hourly emergence of a mechanical bird from a pendulum-driven cuckoo clock to measure the rate at which the earth spins. There is clearly a kind of symmetry here, since we could equally well take our planet’s rotation as the standard and use it to measure the frequency with which the bird pops out of the door. Schematically, let’s represent this measurement process with the following notation, which is part of a system called called birdtracks: 3
\[c \rightarrow e = 24\]
Here \(c\) represents the cuckoo clock and e the rotation of the earth. Although the measurement relationship is nearly symmetric, the arrow has a direction, because, for example, the measurement of the earth’s rotational period in terms of the clock’s frequency is
\[c \rightarrow e = (1\: hr^{-1})(24\: hr) = 24\]
but the clock’s period in terms of the earth’s frequency is
\[e \rightarrow c = \frac{1}{24}\]
We say that the relationship is not symmetric but “dual.” By the way, it doesn’t matter how we arrange these diagrams on the page. The notations \(c \rightarrow e\) and \(e \rightarrow c\) mean exactly the same thing, and expressions like this can even be drawn vertically.
Suppose that \(e\) is a displacement along some one-dimensional line of time, and we want to think of it as the thing being measured. Then we expect that the measurement process represented by \(c\) produces a real-valued result and is a linear function of \(e\). Since the relationship between \(c\) and \(e\) is dual, we expect that \(c\) also belongs to some vector space. For example, vector spaces allow multiplication by a scalar: we could double the frequency of the cuckoo clock by making the bird come out on the half hour as well as on the hour, forming \(2c\). Measurement should be a linear function of both vectors; we say it is “bilinear.”
Duality
The two vectors \(c\) and \(e\) have different units, \(hr^{-1}\) and \(hr\), and inhabit two different one-dimensional vector spaces. The “ﬂavor” of the vector is represented by whether the arrow goes into it or comes out. Just as we used notation like \(\overrightarrow{v}\) in freshman physics to tell vectors apart from scalars, we can employ arrows in the birdtracks notation as part of the notation for the vector, so that instead of writing the two vectors as \(c\) and \(e\), we can notate them as \(c \to\) and \(\to e\). Performing a measurement is like plumbing. We join the two “pipes” in \(c \to\; \to e\) and simplify to \(c \to e\).
A confusing and nonstandardized jungle of notation and terminology has grown up around these concepts. For now, let’s refer to a vector such as \(\to e\), with the arrow coming in, simply as a “vector,” and the type like \(c \to\) as a “covector.” In the one-dimensional example of the earth and the cuckoo clock, the roles played by the two things were completely equivalent, and it didn’t matter which one we expressed as a vector and which as a covector.
References
1 Astronomy and Astrophysics 326 (1997) 924, adsabs.harvard.edu/full/1997A&26A...326..924M
2 arxiv.org/abs/1205.4235
3 The system used in this book follows the one defined by Cvitanovi´c, which was based closely on a graphical notation due to Penrose. For a more complete exposition, see the Wikipedia article “Penrose graphical notation” and Cvitanovi´c’s online book at birdtracks.eu.


2.2: Introduction to No es la gripe, solo son mis alergias
selected template will load here
This action is not available.
In this section, we’ll look at vocabulary dealing with illness and health.
As you learn about medical terminologies you will find that many of the words are similar to English. This is due to the fact that these terms share a connection to a Latin background. Medical terms are usually created by combining prefixes (the beginning of a word) with a suffix (the ending of a word) to create the name of a condition or cure. Most of these prefixes and suffixes have the same meanings across the two languages. For example, take “hypertension/hipertensión.” The prefix “hyper-/hiper-” means “high” or “over” and the suffix “-tension” refers to pressure, meaning a high blood pressure. Therefore, at this point of practice, don’t be afraid to try to make educated guesses!


13.2: Microsoft Access- Main Features and Navigation
By the end of this section, you will be able to:
- Demonstrate navigation of the Microsoft Access screen interface
- Identify the main features of Access
- Understand how to close Access
The benefit of Microsoft Access is that its design and structure enable it to communicate with other systems that hold data. Unlike larger database systems that run on dedicated servers, Access runs on a regular desktop computer. Because it is a desktop database system, the user can work within the database without connecting to those larger systems.
As a Microsoft software package, Access follows many of the conventions of Microsoft Office. This provides a consistent, familiar interface for work and ensures that file transfer is seamless between Microsoft Office software, for example, between Access and Excel.
What Is Access?
Access is a RDBMS that can import and manipulate data from other systems on your desktop computer. While data science professionals work directly with the data warehouse and can create queries using programming languages such as R and Python, individuals who need to analyze data as part of their jobs within a department often find Access to be a user-friendly software package that has the power to access, retrieve, and report data using a Microsoft interface. Access also integrates well with Microsoft Excel , providing even more analytical and reporting power that can inform decision-making.
In small businesses and organizations, Access can act as the main data store; however, it is much more likely that you will use data that is collected and stored in other systems and then imported into Access for analysis.
The Role of Access in an Organization
Access is one of the many tools that make up an organization’s data landscape —the entirety of data storage and sharing in an organization. An organization may have both production database s and analytical database s.
Production databases are systems that handle daily transactions related to an organization’s core business functions, such as sales, inventory, human resources, payroll, and purchasing. Because production databases provide value to the organization, companies devote significant resources to ensure they remain online and active. Companies do not work directly with a production system when analyzing a business because this may interfere with regular operations for customer transactions; instead, companies use a data warehouse or analytical database for building new applications or testing new features. This type of database is entirely separate from the production database, so changes to the analytical database will not modify the production database until the developer is ready to push the changes to the production database after thorough testing.
Analytical databases ( data warehouse s) serve as an organization’s single point of truth (SPOT) in which a primary copy of a data element is stored. Changes made to a data element outside of the data warehouse are not considered final until the primary data element in the data warehouse is also updated. These large databases contain a vast number of records, each of which stores historical information about the organization itself. A data warehouse is not a production system; rather, it is filled with data from the production system, collected over time and organized in a way that makes retrieval and use easier for business use. We can request information from the data warehouse, which is then exported to a file we can use in our analytical software, such as Access or Excel. The ultimate goal of using data is to make good decisions based on the data and not on emotions.
An organization’s data landscape can also contain numerous flat files held within teams and workgroups, other systems that run on databases that should be connected to the data warehouse, and even information from vendors and public sources that can inform data. The data landscape may include information scraped from the web, downloaded and processed from public sources, and environmental data. The extent of an organization’s data landscape is based on the size and role of the company; larger companies will often have more complex data landscapes, as will organizations that are undergoing digital transformation and either have or are building a core platform for all business functions.
Access and Excel
Access and Excel are both software packages in Microsoft Office. They are complementary: Although they have some functions in common, each package is designed for specific uses. Sometimes, the overlap or commonalities between them can make it difficult to know which is most useful. Both Access and Excel can store large amounts of data, run powerful queries, use analysis tools to manipulate data, and perform complex calculations. However, each software has its own strengths, and choosing the right software will make your work easier, as Table 13.2 outlines.
| Use Excel to: | Use Access when you: |
|---|---|
| Analyze a specific dataset | Need to track and record data regularly |
| Perform complex numerical calculations | Need to ensure data integrity |
| Work with a table for calculations | Expect data to be accessed by multiple users |
| Perform what-if analysis to run scenarios | Store data in multiple tables |
| Create advanced pivot tables and charts | Need to connect to other data sources to refresh data (either manually or on a schedule) |
| Create visualizations | Secure data at the user level (each user can have different roles and access to specific data) |
Access and Excel can be used together. It is possible to move data from the data warehouse to Access and then, as necessary, query and export data from Access to Excel for further analysis and visualization. Transferring data from one program to another is one of the benefits of using a desktop database program; as part of the Microsoft suite of products, both Access and Excel can communicate effectively, so users can also import flat files from Excel and convert them to relational tables.
Features of Access
Access is a cost-effective database solution. It runs on a desktop computer and provides an easy-to-use interface that is familiar to users of other Microsoft software. Access interacts well with Excel for deeper analysis, with Word for reporting, and with PowerPoint for presentations. It can work with data from external sources, such as a data warehouse, or with exported files from the web and other systems.
Microsoft Access:
- Creates the framework or structure to store small bits of information (data) in a database
- Allows users to open tables and scroll through records
- Provides an easy-to-use interface for adding, modifying, and retrieving records
- Has a process for creating queries to select, combine, and calculate data from multiple tables
- Provides user-friendly screens and support for creating queries, forms, and reports that allow access and use of the data held in the database
One difference between Access and other Microsoft 365 software is in its introduction or opening screens. Access loads one database at a time, and all screens focus on the open database. When you first open Access, the screen will be blank, Figure 13.6 until you load a database. Once loaded, the screen shows all available commands.
As you move through Access, different commands will become available based on the active objects. For example, you will see different commands when working in tables versus working with queries. The main working area in Access is the center of the screen, with navigation on the left and commands across the top.
The Opening Screen
When Access is open without a database, most commands either do not appear or are unavailable (grayed out). Without a database, the software is inactive. When a database is open on the screen, you will see commands available on the ribbon along the top (Figure 13.7) and a navigation area along the left edge of your screen (Figure 13.8). Only one database can be open at a time in Access, and all software is focused on supporting your open database.
As you click through the ribbon, notice that some commands are not available unless you have specific database objects open, and you can move between objects in your database by using the navigation area on the left side of your database screen.
Although there are many different database software programs in use, most of them have the same parts. Access, as a RDBMS, follows the typical database conventions, or standards, in use in all relational database systems.
You can think of Access as a container that holds database objects, which, recall, are items in a relational database. Each object plays a role in managing the database and the data it stores. Because the database is electronic, it can grow to hold many objects and can create links and new uses for each over time.
Like many databases, Access databases usually contain tables, forms, reports, and queries. (They can also contain macros and modules, which you will learn about in Advanced Database Use.)
Locating and Opening Database Objects
Once you open the database or the container that holds database the object , you will see a navigation area open on the left side of the screen. This area has headings for each type of object that is present in the database, followed by the names of each object you have. To open any object, double-click on its name, and it will open in the main work area on the right side of the screen (Figure 13.9). To close an object, click on the X next to the name of the object.
Tables
Tables hold the data in a database. One database can contain many table s, each of which holds a different record (or set of related information) or parts of a record. One benefit of a relational database is its ability to split large sets of information about a specific item, person, or idea over multiple tables. This action speeds up processing and gives the database a better organizational structure.
As you learned in What Is a Database?, tables are built with rows and columns. The rows of a table consist of individual records or lines of related data. The columns represent fields or headings referring to information that is likely to be included in all the records. This structure, which characterizes all relational databases, makes it possible to share datasets between different database software packages. An example of table structure is presented in Table 13.3.
| ID | FirstName | LastName | MiddleInitial |
|---|---|---|---|
| 1234 | Mary | Nanas | A |
| 2345 | Jose | Garcia | I |
| 3456 | Andre | Williams | C |
| 4567 | Lisa | Estrich | J |
| 5678 | Pat | Johnson | E |
| 6789 | Stevie | Larose | Y |
This small table contains four fields—ID, FirstName, LastName, MiddleInitial—and six records, which contain the information for each field for six different employees.
Queries
Queries are a way of asking questions of a dataset. Recall that a query is a structured request for information, allowing you to choose what information appears, the order in which it appears, and how it is organized. Queries can also be used to calculate information based on fields. Typically, when you interact with the database, you will be creating queries to satisfy business requirements.
Like other database objects, queries appear in the navigation area of the database. To open a query, double-click it, and it will appear in the work area of your screen. In Access, you can change how you view a query, choosing either Datasheet View, which shows the results of the query in table form, or Design View, which shows the structure of the query and allows you to modify it. You will learn more about these views in Querying a Database.
Reports
A report shows the results of a query, formatted in a business-friendly way. Creating a report allows you to choose the precise information you want to include and the format in which you would like your audience to view it. Since tables and queries typically return results in the form of tables, reports allow you to focus on the message your audience will need for decision-making, not on the data. One advantage of creating reports is that you can set up standard reports that you will need to run frequently (e.g., a report of sales totals) and then open them and view the latest results as often as necessary.
Like other objects, reports in Access can be created in a Wizard that walks you through the basics, and then you can adjust the report in Design View. Building a useful report can take time and care to ensure it summarizes and displays information in an actionable way that supports the work group’s goals.
Forms
Recall that a form in a database allows users to work with records in an easy-to-use format. Forms limit the individual’s view of the database to what is necessary for a specific task or information request. On most websites, we interact with forms that are built on the basis of queries: They request and present data taken from the underlying table and query structure. Forms are often based on predefined queries that run over and over again and, as with a search for a product, may prompt the user for a response.
Forms appear in the navigation area of the database and can be opened in Layout or Design View , which allows you to modify the form itself (Figure 13.10), or in Form View (ready for use) (Figure 13.11).
Link to Learning
Sometimes the best way to learn is to explore on your own. Visit Microsoft’s featured Access templates page to choose a template and then download, open, and explore it in Access.
Closing Databases
In Access, only one database can be open at a time. As you work, all on-screen commands will focus on this open database, and you will need to close the database itself to end your working session. When you are ready to close the database, Access will either save objects automatically or prompt you to save objects before closing. In general, Access will automatically save records when you are done with them (i.e., when you leave the row) but will prompt you to save changes to objects, such as tables and queries, so you can decide whether you want to save them.
If you want to make sure an object is saved in Access before you close the database, you can use any typical Save command shortcut (e.g., File>Save, Ctrl+S ) to request that Access save your changes. Access will prompt you if you need to take any additional steps.
When you are done working with your database, click on File>Close to close all objects and exit the Access software. Be sure to close your database every time you are done using it; this will ensure that all objects are closed and that data is correctly written to the records within the database.


13.1: Introduction to the Skeletal System
Learning Objectives
- Apply the rules of medical language to build, analyze, spell, pronounce, abbreviate, and define terms as they relate to the skeletal system
- Identify meanings of key word components of the skeletal system
-
Categorize diagnostic, therapeutic, procedural, or anatomic terms related to the skeletal
system - Use terms related to the skeletal system
- Use terms related to the diseases and disorders of the skeletal system
The skeletal system is composed of bones, cartilage, and ligaments. It performs several functions, including supporting the body, protecting internal organs, producing blood cells, and storing/releasing minerals. [1]
This chapter will review common word components related to the skeletal system to assist learners in analyzing, building, and defining medical terms. Other terms, whose definitions cannot be easily built from word components, will be described in context based on the anatomy and physiology of the skeletal system and common diseases and disorders. Medical specialists, diagnostic tests, and procedures related to the skeletal system will also be discussed.
View a supplementary YouTube video [2] on the skeletal system: The Skeletal System: Crash Course Anatomy & Physiology #19
- This work is a derivative of Anatomy & Physiology by OpenStax and is licensed under CC BY 4.0. Access for free at https://openstax.org/details/books/anatomy-and-physiology-2e ↵
- CrashCourse. (2015, May 18). The skeletal system: Crash Course Anatomy & Physiology #19 [Video]. YouTube. All rights reserved. https://www.youtube.com/watch?v=rDGqkMHPDqE ↵


Skip to main content
-
animism
-
belief that objects are alive and possess consciousness
-
associative play
-
children begin focusing on and engaging with another person while playing, such as by assisting one another
-
autobiographical memory
-
memories of our individual life experiences and events
-
centration
-
tendency for preoperational children to focus on one characteristic of an object or situation at a time
-
code-switching
-
practice of alternating between two or more languages or language varieties within one conversation
-
conservation
-
cognitive ability to understand that changing the appearance of a substance does not alter its mass, number, or volume
-
cooperative play
-
children fully engage in both a play partner and the activity
-
dyspraxia
-
developmental coordination disorder in which children have difficulty executing typical movements
-
egocentrism
-
preoperational children’s tendency to have limited logical reasoning abilities because they perceive things from only their own point of view, according to Piaget’s stage theory of cognitive development
-
false belief
-
children’s belief that other people have the same knowledge they themselves possess
-
fine motor skill
-
precise movements of the hands and fingers
-
food desert
-
geographic area where fresh food is not readily available
-
gross motor skill
-
large body movements of the head, torso, arms, and legs
-
literacy
-
ability to read, write and understand information
-
macronutrients
-
nutrient-rich carbohydrates, proteins, and fats needed for energy
-
night terrors
-
intense fear or terror that occurs during deep sleep
-
onlooker play
-
children observe others at play but do not become physically involved
-
operation
-
logical and reversible mental action
-
parallel play
-
children engage in similar play in close proximity but still play separately
-
pragmatics
-
social cues learned in language
-
preoperational stage
-
stage in which children begin to represent their thoughts and ideas using symbols such as words and images; the second stage in Piaget’s stage theory of cognitive development
-
scaffolding
-
process of providing temporary learning support when tasks are just outside a person’s range of ability
-
semantics
-
a feature of language concerned with meaning and logic
-
sociocultural theory of cognitive development
-
theory of development proposed by Vygotsky that emphasizes the role of social and cultural factors on individual cognitive development
-
solitary play
-
(also, independent play) children play alone focusing on their own activity
-
stunting
-
impaired growth in height, often due to malnutrition
-
symbolic representation
-
representation of thoughts and ideas using symbols, such as words and images
-
syntax
-
rules used in language to construct full sentences
-
theory of mind
-
awareness that our beliefs and perspectives are different from other people’s
-
unoccupied play
-
children observe their environment and perform random movements
-
zone of proximal development (ZPD)
-
range of activities that cannot be completed alone but can be accomplished with the aid of a more skilled adult or peer


4.5: Exponential Relationships (3 of 6)
Learning Objectives
- Use an exponential model (when appropriate) to describe the relationship between two quantitative variables. Interpret the model in context.
Let’s summarize what we have learned about exponential growth models:
The general form of an exponential growth model is y = C · b x .
- C is the initial value . It is the y -value when x = 0. It is also the y -intercept.
- b is the growth factor; it will always be greater than 1 in cases of growth. From the growth factor, we can determine the percentage increase in y for each additional 1 unit increase in x .
Let’s compare the general form of an exponential growth model to the general form for a linear model: y = a + bx .
- In the linear model, a is the initial value . It is the y -value when x = 0. It is also the y -intercept.
- b is the slope . From the slope, we can determine the amount and direction the y -value changes for each additional 1 unit increase in x .
Now we apply what we have learned about exponential growth to find a model for a set of data.
In this activity, you use a simulation to find an exponential model that fits the population growth of Kenya.
Here are the data graphed in the scatterplot in the simulation.
Notice that the Kenyan population growth has a strong positive exponential form. Use the sliders in the simulation to adjust the values of C and b to find a reasonable exponential model that fits this data.
Click here to open this simulation in its own window.
Contributors and Attributions
- Concepts in Statistics. Provided by : Open Learning Initiative. Located at : http://oli.cmu.edu . License : CC BY: Attribution


4.7: Exponential and Logarithmic Models
- Foundational Topics
-
-
Sets and Numbers
- Scientific Notation
-
Sets and Numbers
In this section, you will:
- Model exponential growth and decay.
- Use Newton's Law of Cooling.
- Use logistic-growth models.
- Express an exponential model in base \(e\).
- Use logarithmic models.
Figure \( \PageIndex{ 1 } \): A nuclear research reactor inside the Neely Nuclear Research Center on the Georgia Institute of Technology campus (credit: Georgia Tech Research Institute)
We have already explored some basic applications of exponential and logarithmic functions. In this section, we explore some important applications in more depth, including radioactive isotopes and Newton's Law of Cooling .
Modeling Exponential Growth and Decay
In real-world applications, we need to model the behavior of a function. In mathematical modeling, we choose a familiar general function with properties that suggest that it will model the real-world phenomenon we wish to analyze. In the case of rapid growth, we may choose the exponential growth function:\[P(t) = P_0 e^{k t}, \nonumber \]where \( P_0 \) is equal to the value at time zero, \(e\) is Euler's constant , and \(k\) is a positive constant that determines the rate (percentage) of growth. We may use the exponential growth function in applications involving doubling time , the time it takes for a quantity to double. Such phenomena as wildlife populations, financial investments, biological samples, and natural resources may exhibit growth based on a doubling time. In some applications, however, as we will see when we discuss the logistic equation, the logistic model sometimes fits the data better than the exponential model.
On the other hand, if a quantity is falling rapidly toward zero, without ever reaching zero, then we should probably choose the exponential decay model. Again, we have the form \(P(t) = P_0 e^{k t}\) where \( P_0 \) is the starting value, and \(e\) is Euler's constant. Now \(k\) is a negative constant that determines the rate of decay. We may use the exponential decay model when we are calculating half-life , or the time it takes for a substance to exponentially decay to half of its original quantity. We use half-life in applications involving radioactive isotopes.
In our choice of a function to serve as a mathematical model, we often use data points gathered by careful observation and measurement to construct points on a graph and hope we can recognize the shape of the graph. Exponential growth and decay graphs have a distinctive shape, as we can see in Figure \( \PageIndex{ 2 } \) and Figure \( \PageIndex{ 3 } \). It is important to remember that, although parts of each of the two graphs seem to lie on the \( x \)-axis, they are really a tiny distance above the \( x \)-axis.
Figure \( \PageIndex{ 2 } \): A graph showing exponential growth. The equation is \(y = 2 e^{3 x} \).
Figure \( \PageIndex{ 3 } \): A graph showing exponential decay. The equation is \(y = 3 e^{−2x} \).
Exponential growth and decay often involve very large or very small numbers. To describe these numbers, we often use orders of magnitude. The order of magnitude is the power of ten, when the number is expressed in scientific notation , with one digit to the left of the decimal. For example, the distance to the nearest star, Proxima Centauri, measured in kilometers, is \(40,113,497,200,000\) kilometers. Expressed in scientific notation, this is \(4.01134972 \times 10^{13}\). So, we could describe this number as having order of magnitude \(10^{13}\).
The following note is provided as a reminder of the properties of the graphs of exponential functions.
An exponential function with the form \(P(t) = P_0 e^{k t}\) has the following characteristics:
- one-to-one function
- horizontal asymptote: \(y=0\)
- domain: \((– \infty , \infty )\)
- range: \((0, \infty )\)
- \(x\)-intercept: none
- \(y\)-intercept: \((0,P_0)\)
- increasing if \(k > 0\) (see Figure \( \PageIndex{ 4 } \))
- decreasing if \(k < 0\) (see Figure \( \PageIndex{ 4 } \))
Figure \( \PageIndex{ 4 } \): An exponential function models exponential growth when \(k > 0\) and exponential decay when \(k < 0\).
A population of bacteria doubles every hour. If the culture started with \( 10 \) bacteria, graph the population as a function of time.
- Solution
-
When an amount grows at a fixed percent per unit time, the growth is exponential. To find \( P_0 \) we use the fact that \( P_0 \) is the amount at time zero, so \(P_0 = 10\). To find \(k\), use the fact that after one hour (\(t=1\)) the population doubles from \(10\) to \(20\). The formula is derived as follows\[ \begin{array}{rrclcl} & 20 & = & 10 e^{k \cdot 1} & \quad & \left( \text{substituting} \right) \\[6pt] \implies & 2 & = & e^k & \quad & \left( \text{dividing both sides by }10 \right) \\[6pt] \implies & \ln 2 & = & k & \quad & \left( \text{applying the natural logarithm to both sides} \right) \\[6pt] \end{array} \nonumber \]So \( k = \ln (2)\). Thus, the equation we want to graph is\[y = 10 e^{( \ln 2 )t} = 10 \left( e^{\ln2} \right)^t = 10 \cdot 2^t. \nonumber \]The graph is shown in Figure \( \PageIndex{ 5 } \).
Figure \( \PageIndex{ 5 } \): The graph of \(P(t) = 10 e^{( \ln 2 ) t} \)The population of bacteria after ten hours is \(10,240\). We could describe this amount as being an order of magnitude \(10^4\). The population of bacteria after twenty hours is \(10,485,760\), which is an order of magnitude of \(10^7\), so we could say that the population has increased by three orders of magnitude in ten hours.
Half-Life
In previous sections, we learned the properties and rules for both exponential and logarithmic functions. We have seen that any exponential function can be written as a logarithmic function and vice versa. We have used exponents to solve logarithmic equations and logarithms to solve exponential equations. We are now ready to combine our skills to solve equations that model real-world situations, whether the unknown is in an exponent or in the argument of a logarithm.
One such application is in science, in calculating the time it takes for half of the unstable material in a sample of a radioactive substance to decay, called its half-life . Table \( \PageIndex{ 1 } \) lists the half-life for several of the more common radioactive substances.
| Substance | Use | Half-life |
|---|---|---|
| gallium-67 | nuclear medicine | 80 hours |
| cobalt-60 | manufacturing | 5.3 years |
| technetium-99m | nuclear medicine | 6 hours |
| americium-241 | construction | 432 years |
| carbon-14 | archeological dating | 5,715 years |
| uranium-235 | atomic power | 703,800,000 years |
To find the half-life of a function describing exponential decay, solve the following equation:\[ \dfrac{1}{2} P_0 = P_0 e^{kt}. \nonumber \]We find that the half-life depends only on the constant \(k\) and not on the starting quantity \( P_0 \).
The formula is derived as follows:\[ \begin{array}{rrclcl} & \dfrac{1}{2} P_0 & = & P_0 e^{kt} & & \\[6pt] \implies & \dfrac{1}{2} & = & e^{k t} & \quad & \left( \text{dividing both sides by }P_0 \right) \\[6pt] \implies & \ln \left(\dfrac{1}{2}\right) & = & k t & \quad & \left( \text{taking the natural logarithm of both sides} \right) \\[6pt] \implies & - \dfrac{\ln \left( 1/2 \right)}{k} & = & t & \quad & \left( \text{dividing both sides by }k \right) \\[6pt] \end{array} \nonumber \]Since \(t\), the time, is positive, \(k\) must, as expected, be negative. This gives us the half-life formula\[ t = −\dfrac{\ln(2)}{k}. \nonumber \]While most students tend to memorize this formula, your are expected to understand it and derive it, if needed. In fact, I will always use this derivation in the examples so that you get used to understanding why the formula is true.
The half-life of carbon-14 is 5,730 years. Express the amount of carbon-14 remaining as a function of time, \(t\).
- Solution
-
This formula is derived as follows:\[ \begin{array}{rrclcl} & P(t) & = & P_0 e^{k t} & & \\[6pt] \implies & \dfrac{1}{2} P_0 & = & P_0 e^{k \cdot 5730} & \quad & \left( \text{substituting} \right) \\[6pt] \implies & \dfrac{1}{2} & = & e^{k \cdot 5730} & \quad & \left( \text{dividing both sides by }P_0 \right) \\[6pt] \implies & \ln \left(\dfrac{1}{2}\right) & = & 5730 k & \quad & \left( \text{taking the natural logarithm of both sides} \right) \\[6pt] \implies & \dfrac{\ln \left(1/2\right)}{5730} & = & k & \quad & \left( \text{dividing both sides by }5730 \right) \\[6pt] \end{array} \nonumber \]Substituting this back into the original function, we get\[ P(t) = P_0 e^{\left( \frac{\ln(1/2)}{5730} \right)t}. \nonumber \]We observe that the coefficient of \(t\), \( \frac{ln(0.5)}{5730} \approx −1.2097 \times 10^{−4}\), is negative, as expected in the case of exponential decay.
The half-life of plutonium-244 is 80,000,000 years. Find a function that gives the amount of plutonium-244 remaining as a function of time, measured in years.
Radiocarbon Dating
The formula for radioactive decay is important in radiocarbon dating, which is used to calculate the approximate date a plant or animal died. Radiocarbon dating was discovered in 1949 by Willard Libby, who won a Nobel Prize for his discovery. It compares the difference between the ratio of two isotopes of carbon in an organic artifact or fossil to the ratio of those two isotopes in the air. It is believed to be accurate to within about \( 1\% \) error for plants or animals that died within the last 60,000 years.
Carbon-14 is a radioactive isotope of carbon that has a half-life of 5,730 years. It occurs in small quantities in the carbon dioxide in the air we breathe. Most of the carbon on Earth is carbon-12, which has an atomic weight of 12 and is not radioactive. Scientists have determined the ratio of carbon-14 to carbon-12 in the air for the last 60,000 years, using tree rings and other organic samples of known dates - although the ratio has changed slightly over the centuries.
As long as a plant or animal is alive, the ratio of the two isotopes of carbon in its body is close to the ratio in the atmosphere. When it dies, the carbon-14 in its body decays and is not replaced. By comparing the ratio of carbon-14 to carbon-12 in a decaying sample to the known ratio in the atmosphere, the date the plant or animal died can be approximated.
Since the half-life of carbon-14 is 5,730 years, we derive a formula for the amount present at any time \( t \) as follows:\[ \begin{array}{rrclcl} & P(t) & = & P_0 e^{k t} & & \\[6pt] \implies & \dfrac{1}{2} P_0 & = & P_0 e^{k \cdot 5730} & \quad & \left( \text{after 5,730 years, there is half the initial amount remains} \right) \\[6pt] \implies & \dfrac{1}{2} & = & e^{5730 k} & \quad & \left( \text{dividing both sides by }P_0 \right) \\[6pt] \implies & \ln \left(\frac{1}{2}\right) & = & 5730 k & \quad & \left( \text{taking the natural logarithm of both sides} \right) \\[6pt] \implies & \dfrac{\ln (0.5)}{5730} & = & k & \quad & \left( \text{dividing both sides by }5730 \right) \\[6pt] \end{array} \nonumber \]Thus, the formula for the amount of carbon-14 remaining after \(t\) years is\[P(t) \approx P_0 e^{ \left( \frac{\ln(0.5)}{5730} \right)t}, \nonumber \]where
- \(P(t)\) is the amount of carbon-14 remaining at time \( t \), and
- \( P_0 \) is the amount of carbon-14 when the plant or animal began decaying.
Again, I highly discourage you from memorizing this formula. It's fast to derive, and the half-life changes based on substance given, that it is much preferred to derive the formula each time.
A bone fragment is found that contains 20% of its original carbon-14. To the nearest year, how old is the bone?
- Solution
-
We are told that 20% of an initial sample remains. That is, the current amount of carbon-14 in the sample is \( 0.20 P_0 \). Therefore, we want to solve \( 0.2 P_0 = P_0 e^{k t} \). This gives\[ 0.2 = e^{kt} \implies \ln 0.2 = k t \implies t = \dfrac{\ln 0.2}{k}. \nonumber \]Since we already have computed \( k \) for carbon-14 eariler, we substitute that value into this formula:\[ t = \dfrac{\ln 0.2}{k} = \dfrac{\ln 0.2}{\frac{\ln 0.5}{5730}} = \dfrac{5730 \ln(0.2)}{\ln{0.5}} \approx 13301 \nonumber \]The bone fragment is about 13,301 years old.
In reality, the instruments that measure the percentage of carbon-14 are extremely sensitive and a scientist would need to do much more work than we did in order to be satisfied. Even so, carbon dating is only accurate to about 1%, so the age of the sample from Example \( \PageIndex{ 3 } \) should be given as\[13,301 \text{ years } \pm \text{ 1% of } 13,301 \text{ years } = 13,301 \pm 133 \text{ years}. \nonumber \]
Cesium-137 has a half-life of about 30 years. If we begin with 200 mg of cesium-137, will it take more or less than 230 years until only 1 milligram remains?
Calculating Doubling Time
For decaying quantities, we determined how long it took for half of a substance to decay. For growing quantities, we might want to find out how long it takes for a quantity to double. As we mentioned above, the time it takes for a quantity to double is called the doubling time .
Given the basic exponential growth equation \( P(t) = P_0 e^{kt} \), doubling time can be found by solving for when the original quantity has doubled, that is, by solving \(2 P_0 = P_0 e^{kt} \). Rather than attracting you to memorize a formula, let's show how to derive it while in an example.
According to Moore's Law, the doubling time for the number of transistors that can be put on a computer chip is approximately two years. Give a function that describes this behavior.
- Solution
-
The formula is derived as follows:\[ \begin{array}{rrclcl} & P(t) & = & P_0 e^{kt} & & \\[6pt] \implies & 2 P_0 & = & P_0 e^{k \cdot 2} & \quad & \left( \text{substituting }t = 2 \text{ and }P(2) = 2P_0 \right) \\[6pt] \implies & 2 & = & e^{2k} & \quad & \left( \text{dividing both sides by }P_0 \right) \\[6pt] \implies & \ln 2 & = & 2k & \quad & \left( \text{taking the natural log of both sides} \right) \\[6pt] \implies & \dfrac{\ln 2}{2} & = & k & \quad & \left( \text{dividing both sides by }2 \right) \\[6pt] \end{array} \nonumber \]Thus, the function is\[ P(t) = P_0 e^{(\ln (2)/2) t}.\nonumber \]
Recent data suggests that, as of 2013, the rate of growth predicted by Moore's Law no longer holds. Growth has slowed to a doubling time of approximately three years. Find the new function that takes that longer doubling time into account.
Newton's Law of Cooling
Exponential decay can also be applied to temperature. When a hot object is left in surrounding air that is at a lower temperature, the object's temperature will decrease exponentially, leveling off as it approaches the surrounding air temperature. On a graph of the temperature function, the leveling off will correspond to a horizontal asymptote at the temperature of the surrounding air. Unless the room temperature is zero, this will correspond to a vertical shift of the generic exponential decay function. This translation leads to Newton's Law of Cooling , the scientific formula for temperature as a function of time as an object's temperature is equalized with the ambient temperature.
The temperature of an object, \(T\), in a surrounding medium having temperature \( T_s \) will behave according to the formula\[T(t) = A e^{kt} + T_s, \nonumber \]where
- \(t\) is time,
- \(A\) is the difference between the initial temperature of the object and the surroundings, and
- \(k\) is a constant, the continuous rate of cooling of the object.
A cheesecake is taken out of the oven with an ideal internal temperature of \( 165^{\circ} \text{F}\) and is placed into a \(35^{\circ} \text{F}\) refrigerator. After 10 minutes, the cheesecake has cooled to \(150^{\circ} \text{F}\). If we must wait until the cheesecake has cooled to \(70^{\circ} \text{F}\) before we eat it, how long will we have to wait?
- Solution
-
Because the surrounding air temperature in the refrigerator is 35 degrees, the cheesecake's temperature will decay exponentially toward 35, following the equation\[ T(t) = A e^{kt} +35. \nonumber \]We know the initial temperature was 165, so \(T(0)=165\). Therefore,\[ 165 = A e^{k \cdot 0} +35 \implies 165 = A + 35 \implies A = 130. \nonumber \]We were given another data point, \(T(10)=150\), which we can use to solve for \(k\).\[ \begin{array}{rrclcl} & 150 & = & 130 e^{k \cdot 10} + 35 & \quad & \left( \text{subsituting the point }\left( 10, 150 \right) \right) \\[6pt] \implies & 115 & = & 130 e^{10k} & \quad & \left( \text{subtracting }35\text{ from both sides} \right) \\[6pt] \implies & \dfrac{115}{130} & = & e^{10k} & \quad & \left( \text{dividing both sides by }130 \right) \\[6pt] \implies & \ln\left(\frac{115}{130}\right) & = & 10k & \quad & \left( \text{taking the natural log of both sides} \right) \\[6pt] \implies & \dfrac{\ln\left(\frac{115}{130}\right)}{10} & = & k & \quad & \left( \text{dividing both sides by }10 \right) \\[6pt] \end{array} \nonumber \]Hence, \( k \approx -0.0123 \). This gives us the equation for the cooling of the cheesecake:\[T(t) \approx 130 e^{–0.0123t} +35. \nonumber \]Now we can solve for the time it will take for the temperature to cool to 70 degrees. To keep numbers as exact as possible, we wait until the end to replace \( k \) with our found value.\[ \begin{array}{rrclcl} & 70 & = & 130 e^{kt} + 35 & \quad & \left( \text{substituting }70\text{ for }T(t) \right) \\[6pt] \implies & 35 & = & 130 e^{kt} & \quad & \left( \text{subtracting }35\text{ from both sides} \right) \\[6pt] \implies & \dfrac{35}{130} & = & e^{kt} & \quad & \left( \text{dividing both sides by }130 \right) \\[6pt] \implies & \ln\left(\frac{35}{130}\right) & = & kt & \quad & \left( \text{taking the natural log of both sides} \right) \\[6pt] \implies & \dfrac{\ln\left(35/130\right)}{k} & = & t & \quad & \left( \text{dividing both sides by }k \right) \\[6pt] \implies & \dfrac{\ln\left(35/130\right)}{\frac{\ln\left(115/130\right)}{10}} & = & t & \quad & \left( \text{substituting }k = \dfrac{\ln\left(\frac{115}{130}\right)}{10} \right) \\[6pt] \end{array} \nonumber \]Computing this value, we get \( t \approx 106.68 \). Thus, it will take about 107 minutes, or one hour and 47 minutes, for the cheesecake to cool to \(70^{\circ} \text{F}\).
A pitcher of water at 40 degrees Fahrenheit is placed into a 70 degree room. One hour later, the temperature has risen to 45 degrees. How long will it take for the temperature to rise to 60 degrees?
Using Logistic Growth Models
Exponential growth cannot continue forever. Exponential models, while they may be useful in the short term, tend to fall apart the longer they continue. Consider an aspiring writer who writes a single line on day one and plans to double the number of lines she writes each day for a month. By the end of the month, she must write over 17 billion lines, or one-half-billion pages. It is impractical, if not impossible, for anyone to write that much in such a short period of time. Eventually, an exponential model must begin to approach some limiting value, and then the growth is forced to slow. For this reason, it is often better to use a model with an upper bound instead of an exponential growth model, though the exponential growth model is still useful over a short term, before approaching the limiting value.
The logistic growth model is approximately exponential at first, but it has a reduced rate of growth as the output approaches the model's upper bound, called the carrying capacity . For constants \( a \), \( b \), and \( c \), the logistic growth of a population over time \(t\) is represented by the model\[f(t) = \dfrac{c}{1 + a e^{−bt}}. \nonumber \]The graph in Figure \( \PageIndex{ 6 } \) shows how the growth rate changes over time. The graph increases from left to right, but the growth rate only increases until it reaches its point of maximum growth rate, at which point the rate of increase decreases.
Figure \( \PageIndex{ 6 } \)
The logistic growth model is\[f(t)= \dfrac{c}{1 + a e^{−bt}}, \nonumber \]where
- \( \frac{c}{1+a}\) is the initial value,
- \(c\) is the carrying capacity , or limiting value , and
- \(b\) is a constant determined by the rate of growth.
An influenza epidemic spreads through a population rapidly, at a rate that depends on two factors: The more people who have the flu, the more rapidly it spreads, and also the more uninfected people there are, the more rapidly it spreads. These two factors make the logistic model a good one to study the spread of communicable diseases. And, clearly, there is a maximum value for the number of people infected: the entire population.
For example, at time \(t=0\) there is one person in a community of 1,000 people who has the flu. So, in that community, at most 1,000 people can have the flu. Researchers find that for this particular strain of the flu, the logistic growth constant is \(b = 0.6030\). Estimate the number of people in this community who will have had this flu after ten days. Predict how many people in this community will have had this flu after a long period of time has passed.
- Solution
-
We substitute the given data into the logistic growth model\[f(t) = \dfrac{c}{1 + a e^{−bt}}. \nonumber \]Because at most 1,000 people, the entire population of the community, can get the flu, we know the limiting value is \(c=1000\). To find \(a\), we use the formula that the number of cases at time \(t=0\) is \( \frac{c}{1+a} = 1\), from which it follows that \(a=999\). This model predicts that, after ten days, the number of people who have had the flu is \(f(t)= \frac{1000}{1 + 999 e^{−0.6030t}} \approx 293.8\). Because the actual number must be a whole number (a person has either had the flu or not) we round down to 293. In the long term, the number of people who will contract the flu is the limiting value, \(c=1000\).
Remember that, because we are dealing with a virus, we cannot predict with certainty the number of people infected. The model only approximates the number of people infected and will not give us exact or actual values.
The graph in Figure \( \PageIndex{ 7 } \) gives a good picture of how this model fits the data.
Figure \( \PageIndex{ 7 } \): The graph of \(f(t)= \frac{1000}{1 + 999 e^{−0.6030t}}\).
Using the model in Example \( \PageIndex{ 6 } \), estimate the number of cases of flu on day 15.
Expressing an Exponential Model in Base \(e\)
While powers and logarithms of any base can be used in modeling, the two most common bases are \(10\) and \(e\). In science and mathematics, the base \(e\) is often preferred. We can use Laws of Exponents and Laws of Logarithms to change any base to base \(e\).
Given a model with the form \(y = a b^x \), we change it to the form \(y = A_0 e^{kx} \) using the following steps.
- Rewrite \(y = a b^x\) as \(y = a e^{\ln\left( b^x \right)} \).
- Use the Power Rule of Logarithms to rewrite \( y \) as \( y = a e^{x \ln( b )} = a e^{\ln( b )x} \).
- Note that \( a = A_0 \) and \( k = \ln( b )\) in the equation \( y = A_0 e^{kx} \).
Change the function \(y = 2.5 (3.1)^x \) so that this same function is written in the form \(y = A_0 e^{kx} \).
- Solution
-
The formula is derived as follows:\[ \begin{array}{rclcl} y & = & 2.5(3.1)^x & & \\[6pt] & = & 2.5 e^{\ln\left( 3.1^x \right)} & \quad & \left( \text{Inverse Property of Exponentials and Logarithms} \right) \\[6pt] & = & 2.5 e^{x\ln\left( 3.1 \right)} & \quad & \left( \text{Laws of Logarithms} \right) \\[6pt] & = & 2.5 e^{\ln\left( 3.1 \right) x} & \quad & \left( \text{Commutative Property of Multiplication} \right) \\[6pt] \end{array} \nonumber \]
Change the function \(y = 3(0.5)^x\) to one having \(e\) as the base.
Logarithmic Scales
For quantities that vary greatly in magnitude, a standard scale of measurement is not always effective, and utilizing logarithms can make the values more manageable. For example, if the average distances from the sun to the major bodies in our solar system are listed, you see they vary greatly.
| Planet | Distance (millions of km) |
| Mercury | 58 |
| Venus | 108 |
| Earth | 150 |
| Mars | 228 |
| Jupiter | 779 |
| Saturn | 1430 |
| Uranus | 2880 |
| Neptune | 4500 |
Placed on a linear scale – one with equally spaced values – these values get bunched up (see Figure \( \PageIndex{ 8 } \)).
Figure \( \PageIndex{ 8 } \): Linear scale of astronomical distances in our solar system.
However, computing the logarithm of each value and plotting these new values on a number line results in a more manageable graph, and makes the relative distances more apparent, as seen in Figure \( \PageIndex{ 9 } \) below Table \( \PageIndex{ 3 } \).(It is interesting to note the large gap between Mars and Jupiter on the log number line. The asteroid belt is located there, which scientists believe is a planet that never formed because of the effects of the gravity of Jupiter.)
| Planet | Distance (millions of km) | log(distance) |
| Mercury | 58 | 1.76 |
| Venus | 108 | 2.03 |
| Earth | 150 | 2.18 |
| Mars | 228 | 2.36 |
| Jupiter | 779 | 2.89 |
| Saturn | 1430 | 3.16 |
| Uranus | 2880 | 3.46 |
| Neptune | 4500 | 3.65 |
Figure \( \PageIndex{ 9 } \): Logarithmic scale of astronomical distances in our solar system.
Sometimes, as shown above in Figure \( \PageIndex{ 9 } \), the scale on a logarithmic number line will show the log values, but more commonly the original values are listed as powers of 10, as shown below in Figure \( \PageIndex{ 10 } \).
Figure \( \PageIndex{ 10 } \)
Estimate the value of point \(P\) on the log scale above
- Solution
-
The point \(P\) appears to be half way between \(-2\) and \(-1\) in log value. So if \(V\) is the value of this point,\[ \log (V) \approx -1.5. \nonumber\]Rewriting in exponential form,\[V \approx 10^{-1.5} \approx 0.0316. \nonumber\]
Place the number 6000 on a logarithmic scale.
- Solution
-
Since \(\log (6000) \approx 3.8\), this point would belong on the log scale about here:
Plot the data in the table below on a logarithmic scale (From http://www.epd.gov.hk/epd/noise_educ...1/intro_5.html , retrieved Oct 2, 2010).
| Source of Sound/Noise | Approximate Sound Pressure in \(\mu\) Pa (micro Pascals) |
| Launching of the Space Shuttle | 2000,000,000 |
| Full Symphony Orchestra | 2000,000 |
| Diesel Freight Train at High Speed at 25 m | 200,000 |
| Normal Conversation | 20,000 |
| Soft Whispering at 2 m in Library | 2,000 |
| Unoccupied Broadcast Studio | 200 |
| Softest Sound a human can hear | 20 |
- Answer
-
Notice that on the log scale above Figure \( \PageIndex{ 10 } \), the visual distance on the scale between points \(A\) and \(B\) and between \(C\) and \(D\) is the same. When looking at the values these points correspond to, notice \(B\) is ten times the value of \(A\), and \(D\) is ten times the value of \(C\). A visual (linear) difference between points corresponds to a relative (ratio) change between the corresponding values.
Logarithms are useful for showing these relative changes. For example, comparing $1,000,000 to $10,000, the first is 100 times larger than the second.\[\dfrac{1,000,000}{10,000} = 100 = 10^2.\nonumber\]Likewise, comparing $1000 to $10, the first is 100 times larger than the second.\[\dfrac{1,000}{10} = 100 = 10^2. \nonumber\]When one quantity is roughly ten times larger than another, we say it is one order of magnitude larger. In both cases described above, the first number was two orders of magnitude larger than the second.
Notice that the order of magnitude can be found as the common logarithm of the ratio of the quantities. On the log scale above, \( B \) is one order of magnitude larger than \(A\), and \(D\) is one order of magnitude larger than \(C\).
Given two values \(A\) and \(B\), to determine how many orders of magnitude \(A\) is greater than \(B\), we compute the common logarithm of the ratio of \( A \) to \( B \). That is,\[ \text{Difference in orders of magnitude } = \log\left(\dfrac{A}{B}\right) \nonumber \]
On the log scale in Figure \( \PageIndex{ 10 } \), how many orders of magnitude larger is \(C\) than \(B\)?
- Solution
-
The value \(B\) corresponds to \(10^2 = 100\) and the value \(C\) corresponds to \(10^5 = 100,000\). The relative change is \(\frac{100,000}{100} = 1000 = \frac{10^5}{10^2} = 10^3\). The log of this value is \( 3 \). Therefore, \(C\) is three orders of magnitude greater than \(B\), which can be seen on the log scale by the visual difference between the points on the scale.
Using the table from Checkpoint \( \PageIndex{ 9 } \), what is the difference of order of magnitude between the softest sound a human can hear and the launching of the space shuttle?
- Answer
-
\(\frac{2 \times 10^9}{2 \times 10^1} = 10^8\). The sound pressure in \(\mu\) Pa created by launching the space shuttle is 8 orders of magnitude greater than the sound pressure in \(\mu\) Pa created by the softest sound a human ear can hear.
Earthquakes
An example of a logarithmic scale is the Richter Scale used for earthquakes. While not the only scale for measuring earthquakes, it is the most commonly known. The Richter scale compares the amplitude, \( A \), of the earthquake's seismographic trace with the amplitude, \( A_0 \), of the smallest detectable earthquake.
For an earthquake with seismic trace amplitude \(A\), a measurement from the baseline of the seismograph, the Richter value , or magnitude of the earthquake, is\[M = \log\left(\dfrac{A}{A_0}\right),\nonumber \]where \(A_0 \) is a baseline measure for the seismograph.
If one earthquake has a Richter value of 6.0, and another has a magnitude of 8.0, how much more powerful (in terms of earth movement) is the second earthquake?
- Solution
-
Since the first earthquake has Richter value of 6.0, we can find the amount of earth movement for that quake, which we'll denote \(A_1\). The value of \(A_0\) is not particularity relevant, so we will not replace it with its value.\[ \begin{array}{rrclcl} & 6.0 & = & \log\left( \dfrac{A_1}{A_0} \right) & & \\[6pt] \implies & 10^{6.0} & = & \dfrac{A_1}{A_0} & \quad & \left( \text{rewriting the logarithm in exponential form} \right) \\[6pt] \implies & 10^{6.0} A_0 & = & A_1 & \quad & \left( \text{multiplying both sides by }A_0 \right) \\[6pt] \end{array} \nonumber \]This tells us the first earthquake has about \(10^6\) times more earth movement than the baseline measure.
Doing the same with the second earthquake, \(A_2\), with a magnitude of 8.0,\[ \begin{array}{rrclcl} & 8.0 & = & \log\left( \dfrac{A_2}{A_0} \right) & & \\[6pt] \implies & 10^{8.0} & = & \dfrac{A_2}{A_0} & \quad & \left( \text{rewriting the logarithm in exponential form} \right) \\[6pt] \implies & 10^{8.0} A_0 & = & A_2 & \quad & \left( \text{multiplying both sides by }A_0 \right) \\[6pt] \end{array} \nonumber \]Comparing the earth movement of the second earthquake to the first,\[\dfrac{A_2}{A_1} = \dfrac{10^{8} A_0} {10^6 A_0} = 10^2 = 100.\nonumber\]The second value's earth movement is 100 times as large as the first earthquake.
One earthquake has magnitude of 3.0. If a second earthquake has twice as much earth movement as the first earthquake, find the magnitude of the second quake.
- Solution
-
Since the first quake has magnitude 3.0,\[3.0 = \log \left( \dfrac{A}{A_0} \right). \nonumber\]Solving for \(A\),\[ \begin{array}{rrclcl} & 3.0 & = & \log\left( \dfrac{A}{A_0} \right) & & \\[6pt] \implies & 10^{3.0} & = & \dfrac{A}{A_0} & \quad & \left( \text{rewriting in exponential form} \right) \\[6pt] \implies & 10^{3.0} A_0 & = & A & \quad & \left( \text{multiplying both sides by }A_0 \right) \\[6pt] \end{array} \nonumber \]Since the second earthquake has twice as much earth movement, for the second quake we get\[S = 2 \cdot 10^{3.0} S_0. \nonumber\]Finding the magnitude,\[ \begin{array}{rcl} M & = & \log\left( \dfrac{2 \cdot 10^{3.0} A_0}{A_0} \right) \\[6pt] & = & \log\left( 2 \cdot 10^{3.0} \right) \\[6pt] & \approx & 3.3010 \\[6pt] \end{array} \nonumber \]The second earthquake with twice as much earth movement will have a magnitude of about 3.3.
In fact, using the Laws of Logarithms, we could show that whenever the earth movement doubles, the Richter magnitude will increase by about 0.301:\[ \begin{array}{rcl} M & = & \log\left( \dfrac{2A}{A_0} \right) \\[6pt] & = & \log\left( 2 \cdot \dfrac{A}{A_0} \right) \\[6pt] & = & \log\left( 2 \right) + \log \left( \dfrac{A}{A_0} \right) \\[6pt] & \approx & 0.3010 + \log \left( \dfrac{A}{A_0} \right) \\[6pt] \end{array} \nonumber \]This illustrates the most important feature of a log scale: that multiplying the quantity being considered will add to the scale value, and vice versa.
Logarithms are useful in measuring the intensities of many quantities based on senses. Earthquakes (the Richter scale ), sound ( decibels ) and acids and bases ( pH ) are common examples. We now present yet a different use of the basic logarithm function, password strength .
The information entropy \(H\), in bits, of a randomly generated password consisting of \(L\) characters is given by\[H = L \log_{2}(N),\nonumber\]where \(N\) is the number of possible symbols for each character in the password. In general, the higher the entropy, the stronger the password.
- If a \(7\) character case-sensitive password is comprised of letters and numbers only, find the associated information entropy.
- How many possible symbol options per character are required to produce a \(7\) character password with an information entropy of \(50\) bits?
- Solutions
-
- There are \(26\) letters in the alphabet, \(52\) if upper and lower case letters are counted as different. There are \(10\) digits (\(0\) through \(9\)) for a total of \(N=62\) symbols. Since the password is to be \(7\) characters long, \(L = 7\). Thus, \(H = 7 \log_{2}(62) = \frac{7 \ln(62)}{\ln(2)} \approx 41.68\).
- We have \(L = 7\) and \(H=50\) and we need to find \(N\). Solving the equation \(50 = 7 \log_{2}(N)\) gives \(N = 2^{50/7} \approx 141.323\), so we would need \(142\) different symbols to choose from.
Chemical systems known as buffer solutions can adjust to small changes in acidity to maintain a range of pH values. Buffer solutions have a wide variety of applications, from maintaining a healthy fish tank to regulating the pH levels in blood.
Blood is a buffer solution. When carbon dioxide is absorbed into the bloodstream, it produces carbonic acid and lowers the pH. The body compensates by producing bicarbonate, a weak base, to partially neutralize the acid. The equation which models blood pH in this situation is\[\text{pH} = 6.1 + \log\left(\dfrac{800}{x} \right),\nonumber\]where \(x\) is the partial pressure of carbon dioxide in arterial blood, measured in torr. Find the partial pressure of carbon dioxide in arterial blood if the pH is \(7.4\).
- Solution
-
We set \(\text{pH} = 7.4\) and get \(7.4 = 6.1 + \log\left(\frac{800}{x} \right)\), or \(\log\left(\frac{800}{x} \right) = 1.3\). Solving, we find \(x = \frac{800}{10^{1.3}} \approx 40.09\). Hence, the partial pressure of carbon dioxide in the blood is about \(40\) torr.
Access these online resources for additional instruction and practice with exponential and logarithmic models.


5.6: Notes
-
- Last updated
- Save as PDF
- Vicki Mayer
- University of California Press
1 Susan Christopherson and Jennifer Clark, Remaking Regional Economies: Power, Labor, and Firm Strategies in the Knowledge Economy (New York: Routledge, 2007).
2 See, for example, Susan Christopherson, “Behind the Scenes: How Transnational Firms Are Constructing a New International Division of Labor in Media Work,” in Cross-Border Cultural Production: Economic Runaway or Globalization? , edited by Janet Wasko (New York: Cambria Press, 2008); Tom O’Regan and Ben Goldsmith, The Film Studio: Film Production in the Global Economy (Lanham, MD: Rowman & Littlefield, 2005); Greg Elmer and Mike Gasher, Contracting Out Hollywood: Runaway Productions and Foreign Location Shooting (Lanham, MD: Rowman & Littlefield, 2005); Mike Gasher, Hollywood North: The Feature Film Industry in British Columbia (Seattle: University of Washington Press, 2002); Toby Miller, Nitin Govil, John McMurria, Richard Maxwell, and Ting Wang, Global Hollywood 2 (London: British Film Institute, 2008); Michael Storper and Susan Christopherson, “Flexible Specialization and Regional Industrial Agglomeration: The Case of the U.S. Motion Picture Industry,” Annals of the Association of American Geographers 77 (1987): 1104–1117; Janet Wasko, How Hollywood Works (London: Sage, 2003).
3 Mayor’s Office of Cultural Economy, 2014 New Orleans Cultural Economy Snapshot , New Orleans, LA, located at www.nola.gov/getattachment/Cultural-Economy/2014-CE-Snapshot-Electronic.pdf/.
4 Adriana Lopez, “A Look into America’s Fastest Growing City,” Forbes.com , July 26, 2012, www. forbes.com/sites/adrianalopez/2012/07/26/a-look-into-americas-fastest-growing-city/.
5 Dave Walker, “On the HBO ‘Treme’ Trail: David Simon, Eric Overmyer Discuss Creation of Prospective Drama,” Nola.com , April 4, 2009, http://blog.nola.com/davewalker/2009/04/on_the_treme_ trail_david_simon.html.
6 Mark Banks, “Moral Economies and Cultural Work,” Sociology 40 (2006): 455–472.
7 Vincanne Adams, Markets of Sorrow, Labors of Faith: New Orleans in the Wake of Katrina (Durham, NC: Duke University Press, 2013).
8 Laurie Ouelette and James Hay, Better Living through Reality TV: Television and Post-Welfare Citizenship (London: Wiley, 2008).
9 Amanda Lagerkvist, “9.11 in Sweden: Commemoration at Electronic Sites of Memory,” Television & New Media 15 (2014): 350–370.
10 See, for example, Melissa Gregg, Work’s Intimacy (London: Polity, 2012).


Skip to main content
-
On behalf of the King, Don Fernando, and of Doña Juana I, his daughter, Queen of Castille and León, subduers of the barbarous nations, we their servants notify and make known to you, as best we can, that the Lord our God, Living and Eternal, created the Heaven and the Earth, and one man and one woman, of whom you and we, all the men of the world at the time, were and are descendants, and all those who came after and before us. But, on account of the multitude which has sprung from this man and woman in the five thousand years since the world was created, it was necessary that some men should go one way and some another, and that they should be divided into many kingdoms and provinces, for in one alone they could not be sustained.
-
Of all these nations God our Lord gave charge to one man, called St. Peter, that he should be Lord and Superior of all the men in the world, that all should obey him, and that he should be the head of the whole Human Race, wherever men should live, and under whatever law, sect, or belief they should be; and he gave him the world for his kingdom and jurisdiction.
-
And he commanded him to place his seat in Rome, as the spot most fitting to rule the world from; but also he permitted him to have his seat in any other part of the world, and to judge and govern all Christians, Moors, Jews, Gentiles, and all other Sects. This man was called Pope, as if to say, Admirable Great Father and Governor of men. The men who lived in that time obeyed that St. Peter, and took him for Lord, King, and Superior of the universe; so also they have regarded the others who after him have been elected to the pontificate, and so has it been continued even till now, and will continue till the end of the world.
-
One of these Pontiffs, who succeeded that St. Peter as Lord of the world, in the dignity and seat which I have before mentioned, made donation of these isles and Tierra-firme to the aforesaid King and Queen and to their successors, our lords, with all that there are in these territories, as is contained in certain writings which passed upon the subject as aforesaid, which you can see if you wish.
-
So their Highnesses are kings and lords of these islands and land of Tierra-firme by virtue of this donation: and some islands, and indeed almost all those to whom this has been notified, have received and served their Highnesses, as lords and kings, in the way that subjects ought to do, with good will, without any resistance, immediately, without delay, when they were informed of the aforesaid facts. And also they received and obeyed the priests whom their Highnesses sent to preach to them and to teach them our Holy Faith; and all these, of their own free will, without any reward or condition, have become Christians, and are so, and their Highnesses have joyfully and benignantly received them, and also have commanded them to be treated as their subjects and vassals; and you too are held and obliged to do the same. Wherefore, as best we can, we ask and require you that you consider what we have said to you, and that you take the time that shall be necessary to understand and deliberate upon it, and that you acknowledge the Church as the Ruler and Superior of the whole world, and the high priest called Pope, and in his name the King and Queen Doña Juana our lords, in his place, as superiors and lords and kings of these islands and this Tierra-firme by virtue of the said donation, and that you consent and give place that these religious fathers should declare and preach to you the aforesaid.
-
If you do so, you will do well, and that which you are obliged to do to their Highnesses, and we in their name shall receive you in all love and charity, and shall leave you, your wives, and your children, and your lands, free without servitude, that you may do with them and with yourselves freely that which you like and think best, and they shall not compel you to turn Christians, unless you yourselves, when informed of the truth, should wish to be converted to our Holy Catholic Faith, as almost all the inhabitants of the rest of the islands have done.
-
But, if you do not do this, and maliciously make delay in it, I certify to you that, with the help of God, we shall powerfully enter into your country, and shall make war against you in all ways and manners that we can, and shall subject you to the yoke and obedience of the Church and of their Highnesses; we shall take you and your wives and your children, and shall make slaves of them, and as such shall sell and dispose of them as their Highnesses may command; and we shall take away your goods, and shall do you all the mischief and damage that we can, as to vassals who do not obey, and refuse to receive their lord, and resist and contradict him; and we protest that the deaths and losses which shall accrue from this are your fault, and not that of their Highnesses, or ours, nor of these cavaliers who come with us. And that we have said this to you and made this Requisition, we request the notary here present to give us his testimony in writing, and we ask the rest who are present that they should be witnesses of this Requisition.”