Fitting an Exponential Function through Two Points
Doubling Time
Half-Life
Annuities and Amortization
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 4.5
4.6Chapter Summary and Review
Key Concepts
Chapter 4 Review Problems
4.7Projects for Chapter 4
5Logarithmic Functions
5.1Inverse Functions
Introduction
Finding a Formula for the Inverse Function
Inverse Function Notation
Graph of the Inverse Function
When Is the Inverse a Function?
Mathematical Properties of the Inverse Function
Symmetry
Domain and Range
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 5.1
5.2Logarithmic Functions
Logarithms and Exponents
Inverse of the Exponential Function
Graphs of Logarithmic Functions
Modeling with Logarithmic Functions
Logarithmic Equations
More About Inverse Functions
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 5.2
Investigation
5.3The Natural Base
The Natural Exponential Function
The Natural Logarithmic Function
Properties of the Natural Logarithm
Solving Equations
Exponential Growth and Decay
Continuous Compounding
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 5.3
5.4Logarithmic Scales
Introduction
Using Log Scales
Equal Increments on a Log Scale
Acidity and the pH Scale
Decibels
The Richter Scale
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 5.4
5.5Chapter Summary and Review
Key Concepts
Chapter 5 Review Problems
5.6Projects for Chapter 5
6Quadratic Functions
6.1Factors and \(x\)-Intercepts
Zero-Factor Principle
Solving Quadratic Equations by Factoring
Applications
Solutions of Quadratic Equations
Equations Quadratic in Form
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 6.1
6.2Solving Quadratic Equations
Squares of Binomials
Solving Quadratic Equations by Completing the Square
The General Case
Quadratic Formula
Applications
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 6.2
6.3Graphing Parabolas
Introduction
The Graph of \(y = ax^2\)
The Graph of \(y= x^2 + c\)
The Graph of \(y = ax^2 + bx\)
Finding the Vertex
The Graph of \(y = ax^2 + bx + c\)
Number of \(x\)-Intercepts
Sketching a Parabola
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 6.3
6.4Problem Solving
Maximum or Minimum Values
The Vertex Form for a Parabola
Graphing with the Vertex Form
Systems Involving Quadratic Equations
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 6.4
6.5Quadratic Inequalities
Solving Inequalities Graphically
Solving Quadratic Inequalities Algebraically
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 6.5
6.6Curve Fitting
Introduction
Finding a Quadratic Function through Three Points
Finding an Equation in Vertex Form
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 6.6
6.7Chapter Summary and Review
Key Concepts
Chapter 6 Review Problems
6.8Projects for Chapter 6
7Polynomial and Rational Functions
7.1Polynomial Functions
Introduction
Products of Polynomials
Special Products
Factoring Cubics
Modeling with Polynomials
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 7.1
7.2Graphing Polynomial Functions
Classifying Polynomials by Degree
Cubic Polynomials
Quartic Polynomials
\(x\)-Intercepts and the Factor Theorem
Zeros of Multiplicity Two or Three
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 7.2
7.3Complex Numbers
Introduction
Imaginary Numbers
Complex Numbers
Arithmetic of Complex Numbers
Products of Complex Numbers
Quotients of Complex Numbers
Zeros of Polynomials
Graphing Complex Numbers
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 7.3
7.4Graphing Rational Functions
Introduction
Domain of a Rational Function
Vertical Asymptotes
Horizontal Asymptotes
Applications
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 7.4
7.5Equations That Include Algebraic Fractions
Solving Equations with Fractions Algebraically
Extraneous Solutions
Formulas
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 7.5
7.6Chapter Summary and Review
Key Concepts
Chapter 7 Review Problems
7.7Projects for Chapter 7
8Linear Systems
8.1Systems of Linear Equations in Two Variables
Solving Systems by Graphing
Solving Systems Algebraically
Inconsistent and Dependent Systems
Applications
An Application from Economics
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 8.1
8.2Systems of Linear Equations in Three Variables
\(3\times 3\) Linear Systems
Back-Substitution
Gaussian Reduction
Inconsistent and Dependent Systems
Applications
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 8.2
8.3Solving Linear Systems Using Matrices
Coefficient Matrix and Augmented Matrix of a System
Elementary Row Operations
Matrix Reduction
Reducing a \(3\times 3\) Matrix
Solving Larger Systems
Reduced Row Echelon Form
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 8.3
8.4Linear Inequalities
Graphs of Inequalities in Two Variables
Linear Inequalities
Using a Test Point
Systems of Inequalities
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 8.4
8.5Linear Programming
The Objective Function and Constraints
Feasible Solutions
The Optimum Solutions
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 8.5
8.6Chapter Summary and Review
Key Concepts
Chapter 8 Review Problems
8.7Projects for Chapter 8
9Sequences and Series
9.1Sequences
Definitions and Notation
Applications of Sequences
Recursively Defined Sequences
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 9.1
9.2Arithmetic and Geometric Sequences
Arithmetic Sequences
The General Term of an Arithmetic Sequence
Geometric Sequences
The General Term of a Geometric Sequence
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 9.2
9.3Series
Introduction
Arithmetic Series
Geometric Series
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 9.3
9.4Infinite Geometric Series
Summation Notation
Infinite Series
Repeating Decimals
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 9.4
9.5The Binomial Expansion
Powers of Other Binomials
The Binomial Coefficient
Using Pascal’s Triangle
Factorial Notation
The Binomial Coefficient in Factorial Notation
The Binomial Theorem
Section Summary
Vocabulary
CONCEPTS
STUDY QUESTIONS
SKILLS
Homework 9.5
9.6Chapter Summary and Review
Key Concepts
Chapter 9 Review Problems
9.7Projects for Chapter 9
Appendices
AAlgebra Skills Refresher
A.1Numbers and Operations
Order of Operations
Parentheses and Fraction Bars
Radicals
Scientific Notation
Section Summary
Vocabulary
SKILLS
Exercises A.1
A.2Linear Equations and Inequalities
Solving Linear Equations
Formulas
Linear Inequalities
Interval Notation
Section Summary
Vocabulary
SKILLS
Exercises A.2
A.3Algebraic Expressions and Problem Solving
Problem Solving
Supply and Demand
Percent Problems
Weighted Averages
Section Summary
Vocabulary
SKILLS
Exercises A.3
A.4Graphs and Equations
Reading a Graph
Graphs of Equations
Section Summary
Vocabulary
SKILLS
Exercises A.4
A.5Linear Systems in Two Variables
Solving Systems by Substitution
Solving Systems by Elimination
Section Summary
Vocabulary
SKILLS
Exercises A.5
A.6Laws of Exponents
Product of Powers
Quotients of Powers
Power of a Power
Power of a Product
Power of a Quotient
Section Summary
Vocabulary
SKILLS
Exercises A.6
A.7Polynomials and Factoring
Polynomials
Products of Polynomials
Products of Binomials
Factoring
Common Factors
Opposite of a Binomial
Polynomial Division
Section Summary
Vocabulary
SKILLS
Exercises A.7
A.8Factoring Quadratic Trinomials
Special Products and Factors
Section Summary
Vocabulary
SKILLS
Exercises A.8
A.9Working with Algebraic Fractions
Reducing Fractions
Products of Fractions
Quotients of Fractions
Sums and Differences of Like Fractions
Lowest Common Denominator
Building Fractions
Sums and Differences of Unlike Fractions
Complex Fractions
Negative Exponents
Section Summary
Vocabulary
SKILLS
Exercises A.9
A.10Working with Radicals
Properties of Radicals
Simplifying Radicals
Sums and Differences of Radicals
Products of Radicals
Rationalizing the Denominator
Simplifying \(\sqrt[n]{x^n} \)
Extraneous Solutions to Radical Equations
Equations with More than One Radical
Section Summary
Vocabulary
SKILLS
Exercises A.10
A.11Facts from Geometry
Right Triangles and the Pythagorean Theorem
Isosceles and Equilateral Triangles
The Triangle Inequality
Similar Triangles
Volume and Surface Area
The Distance Formula
The Midpoint Formula
Circles
Section Summary
Vocabulary
SKILLS
Exercises A.11
A.12Properties of Lines
Horizontal and Vertical Lines
Parallel and Perpendicular Lines
Applications to Geometry
Section Summary
Vocabulary
SKILLS
Reading Questions
Exercises A.12
A.13The Real Number System
Subsets of the Real Numbers
Rational Numbers
Irrational Numbers
Properties of the Real Numbers
Order Properties of the Real Numbers
Section Summary
Vocabulary
SKILLS
Exercises A.13
BUsing a Graphing Calculator
B.1Getting Started
On and Off
Numbers and Operations
Clear and Delete
B.2Entering Expressions
Parentheses
Exponents and Powers
Roots
Absolute Value
Scientific Notation
B.3Editing Expressions
Overwriting
Recalling an Entry
Inserting a Character
Recalling an Answer
B.4Graphing an Equation
Standard Window
Tracing
Multiple Graphs
Setting the Window
Intersect Feature
Other Windows
B.5Making a Table
Using the Auto Option
Using the Ask Option
B.6Regression
Making a Scatterplot
Finding a Regression Equation
Graphing the Regression Equation
B.7Function Notation and Transformation of Graphs
Function Notation
Transformation of Graphs
Translations
Vertical Scalings and Reflections
CUsing a GeoGebra Calculator App
C.1Getting Started
On and Off
Numbers and Operations
Delete and Undo
C.2Entering Expressions
Parentheses
Fractions
Exponents and Powers
Square Roots
Other Roots
Absolute Value
Scientific Notation
Editing an Entry
C.3Graphing an Equation
A Basic Graph with intercepts
Translate and Zoom
Graphing a Function, Making a Table, and Zooming One Axis
C.4More graphing
Finding a Suitable Graphing Window
Multiple Graphs and the Intersect Feature
C.5Regression
Making a Scatterplot and Finding the Regression Line
C.6Troubleshooting the GeoGebra App
DGlossary
EProperties of Numbers
FGeometry formulas
GAnswers to Selected Exercises
HGNU Free Documentation License
Bibliography
Index
Colophon
Section5.4Logarithmic Scales
SubsectionIntroduction
Because logarithmic functions grow very slowly, they are useful for modeling phenomena that take on a very wide range of values. For example, biologists study how metabolic functions such as heart rate are related to an animal’s weight, or mass. The table shows the mass in kilograms of several mammals.
Animal
Shrew
Cat
Wolf
Horse
Elephant
Whale
Mass, kg
\(0.004\)
\(4\)
\(80\)
\(300\)
\(5400\)
\(70,000\)
Imagine trying to scale the \(x\)-axis to show all of these values. If we set tick marks at intervals of \(10,000\) kg, as shown below, we can plot the mass of the whale, and maybe the elephant, but the dots for the smaller animals will be indistinguishable. On the other hand, we can plot the mass of the cat if we set tick marks at intervals of \(1\) kg, but the axis will have to be extremely long to include even the wolf. We cannot show the masses of all these animals on the same scale To get around this problem, we’ll compute the the log of each mass, and use the logs on a new scale. The table below shows the base 10 log of each animal’s mass, rounded to \(2\) decimal places.
Animal
Shrew
Cat
Wolf
Horse
Elephant
Whale
Mass, kg
\(0.004\)
\(4\)
\(80\)
\(300\)
\(5400\)
\(70,000\)
Log (mass)
\(-2.40\)
\(0.60\)
\(1.90\)
\(2.48\)
\(3.73\)
\(4.85\)
The logs of the masses range from \(-2.40\) to \(4.85\text{.}\) We can easily plot these values on a single scale, as shown below. We’d need to keep in mind that we are plotting the logs of the animals’ masses, and not the actual masses. However, remember that a logarithm is really an exponent! For example, the mass of the horse is 300 kg, and \begin{equation*} \text{since} ~~~\log_{10}(300) = {\blert{2.48}}, ~~~\text{then} ~~~ 10^{\blert{2.48}} = 300 \end{equation*} So instead of plotting the logs from the table, we will plot powers of 10 that give the actual masses of the animals, like this: Compare this new scale to the previous one. It looks almost the same, except that the number line is labeled with powers of 10. Even though we computed the log of each mass, we still plotted the actual mass of each animal, in its form as a power of 10. It is the scale on the number line that has changed. A scale labeled with powers of 10 is called a logarithmic scale, or log scale. The powers of 10 on a log scale are evenly spaced, so that the actual values at the tick marks look like this. We can see right away that the increments between tick marks on a log scale are not equal, as they are on a usual linear scale. The increments get larger as we move from left to right on the scale. However, when we are plotting powers of 10 we use the exponents to place the data points on the scale. For example, you can check that the mass of the horse, at \(10^{2.48} = 300\) kg, is plotted about half-way between \(10^2 = 100\) and \(10^3 = 1000\) on the log scale, because 2.48 is about half-way between 2 and 3. Similarly, the mass of the cat, at \(10^{0.60} = 4\) kg, is plotted between \(10^0 = 1\) and \(10^1 = 10\) on the log scale.
Example5.90.
Plot the values on a log scale.
\(x\)
\(0.0007\)
\(0.2\)
\(3.5\)
\(1600\)
\(72,000\)
\(4 \times 10^8\)
Solution.
We first compute the base \(10\) logarithm of each number.
\(x\)
\(0.0007\)
\(0.2\)
\(3.5\)
\(1600\)
\(72,000\)
\(4 \times 10^8\)
\(\log x\)
\(-3.15\)
\(-0.70\)
\(0.54\)
\(3.20\)
\(4.86\)
\(8.60\)
Thus, for example, we see that \(0.0007 = 10^{-3.15}\text{.}\) Then we use the logs to plot each number as a power of 10, estimating its position between integer powers of 10. For example, we plot the first value, \(10^{-3.15}\text{,}\) closer to \(10^{-3}\) than to \(10^{-4}\text{.}\) The finished plot is shown below.
Checkpoint5.91.QuickCheck 1.
A value of 5682.7 would be plotted between which two integers on a log scale?
5682 and 5683
5000 and 6000
5 and 6
3 and 4
Checkpoint5.92.Practice 1.
Complete the table by estimating the logarithm of each point plotted on the log scale below. Then use a calculator to give a decimal value for each point.
\(\log x\)
\(\hphantom{0000}\)
\(\hphantom{0000}\)
\(\hphantom{0000}\)
\(\hphantom{0000}\)
\(x\)
Solution.
\(\log x\)
\(-4\)
\(-2.5\)
\(1.5\)
\(4.25\)
\(x\)
\(0.0001\)
\(0.00316\)
\(31.6\)
\(17,782.8\)
Checkpoint5.93.QuickCheck 2.
What is a log scale used for?
To find the logarithm of a number.
To plot data that covers a wide range of values.
To highlight the curvature in the graph of an exponential function.
To convert logarithms to base 10.
SubsectionUsing Log Scales
By now, you have noticed that the values represented by points on a log scale increase rapidly as we move to the right along the scale. Also notice that \(10^0 = 1\text{,}\) so the "middle" of a log scale represents \(1\) (not zero, as on a linear scale). Points to the left of \(10^0\) represent fractions between \(0\) and \(1\text{,}\) because powers of \(10\) with negative exponents are numbers less than \(1\text{.}\) Their values decrease toward \(0\) as we move to the left, but they never become negative. We cannot plot negative numbers or zero on a log scale, because the log of a negative number or zero is undefined.
Example5.94.
The figure shows a timeline for life on Earth, in units of Mya (million years ago). Approximately how long ago did each of the following events occur?
Formation of Earth
Dinosaurs became extinct
The last ice age
The Crusades
Solution.
We read from the timeline that the Earth was formed between \(10^3\) and \(10^4\text{,}\) or between \(1000\) and \(10,000\) million years ago. We estimate that Earth formed \(5000\) million years ago.
The extinction of the dinosaurs is plotted between \(10^1\) and \(10^2\text{,}\) or between \(10\) and \(100\) million years ago. Because the point is closer to \(10^2\text{,}\) we estimate their extinction at \(70\) million years ago.
The last ice age is plotted just after \(10^{-2}\text{,}\) or \(0.01\) million years ago. One-hundredth of a million is \(10,000\text{,}\) so we estimate that the ice age occurred a little more than \(10,000\) years ago.
The Crusades occurred about \(10^{-3}\text{,}\) or about \(0.001\) million years ago. One-thousandth of a million is \(1000\text{,}\) so the Crusades occurred about \(1000\) years ago, or about \(1000\) A.D.
Checkpoint5.95.Practice 2.
Plot the following dollar values on a log scale.
Postage stamp
\(0.47\)
Notebook computer
\(679\)
One year at Harvard
\(88,600\)
2016 Lamborghini
\(530,075\)
Kobe Bryant salary
\(25,000,000\)
Bill Gates financial worth
\(79,400,000,000\)
U.S. National debt
\(19,341,810,000\)
Solution.
Checkpoint5.96.Pause and Reflect.
How does a log scale differ from a linear scale?
SubsectionEqual Increments on a Log Scale
Log scales allow us to plot a wide range of values, but there is a trade-off. Equal increments on a log scale do not correspond to equal differences in value, as they do on a linear scale. You can see this more clearly if we label the tick marks with their integer values, as well as powers of 10. The difference between \(10^1\) and \(10^0\) is \(10 - 1 = 9\text{,}\) but the difference between \(10^2\) and \(10^1\) is \(100 - 10 = 90\text{.}\) If we include tick marks for intermediate values on the log scale, they look like this. Once again, the difference between, say, \(10^{0.1}\) and \(10^{0.2}\) is not the same as the difference between \(10^{0.2}\) and \(10^{0.3}\text{.}\) The decimal values of the powers \(10^{0.1}\) through \(10^{0.9}\text{,}\) rounded to two places, are shown below. As we move from left to right on this scale, we multiply the value at the previous tick mark by \(10^{0.1}\text{,}\) or about \(1.258\text{.}\) For example, \begin{equation*} \begin{aligned}[t] 10^{0.2} \amp = 1.258 \times 10^{0.1} = 1.585\\ 10^{0.3} \amp = 1.585 \times 10^{0.1} = 1.995 \end{aligned} \end{equation*} and so on. Moving up by equal increments on a log scale does not add equal amounts to the values plotted; it multiplies the values by equal factors.
Checkpoint5.97.QuickCheck 3.
Which statement is false?
We use log scales to graph a variable that has a wide range of values.
On a log scale, we actually plot exponents.
Values less than one appear as negative numbers on a log scale.
Equal increments on a log scale correspond to equal differences in value.
Example5.98.
What number is halfway between \(10\) and \(100\) on a log scale?
Solution.
On a log scale, the number \(10^{1.5}\) is halfway between \(10^1\) and \(10^2\text{,}\) as shown below. Now, \(10^{1.5} = 10\sqrt{10}\text{,}\) or approximately \(31.62\text{.}\) Note how equal increments of \(0.5\) on the log scale correspond to equal factors of \(10^{0.5}\) in the values plotted: \begin{equation*} \begin{aligned}[t] 10 \times 3.162 \amp = 31.62~~~~ \text{ and }~~~~ 31.62 \times 3.162 = 100\\ 10^1 \times 10^{0.5} \amp = 101.5~~~~ \text{ and }~~~~ 101.5 \times 10^{0.5} = 10^2 \end{aligned} \end{equation*}
Checkpoint5.99.Practice 3.
What number is halfway between \(10^{1.5}\) and \(10^2\) on a log scale? Solution.\(56.23\)If we would like to label the log scale with integers, we get a very different-looking scale, one in which the tick marks are not evenly spaced.
Example5.100.
Plot the integer values 2 through 9 and 20 through 90 on a log scale.
Solution.
We compute the logarithm of each integer value.
\(x\)
\(2\)
\(3\)
\(4\)
\(5\)
\(6\)
\(7\)
\(8\)
\(9\)
\(\log x\)
\(0.301\)
\(0.477\)
\(0.602\)
\(0.699\)
\(0.778\)
\(0.845\)
\(0.903\)
\(0.954\)
\(x\)
\(20\)
\(30\)
\(40\)
\(50\)
\(60\)
\(70\)
\(80\)
\(90\)
\(\log x\)
\(1.301\)
\(1.477\)
\(1.602\)
\(1.699\)
\(1.778\)
\(1.845\)
\(1.903\)
\(1.954\)
We plot on a log scale, as shown below. On the log scale in Example 5.100, notice how the integer values are spaced: They get closer together as they approach the next power of \(10\text{.}\) You will often see log scales labeled not with powers of \(10\text{,}\) but with integer values, like this: In fact, log-log graph paper scales both axes with logarithmic scales.
Checkpoint5.101.Practice 4.
The opening page of Chapter 3 shows the "mouse-to-elephant" curve, a graph of the metabolic rate of mammals as a function of their mass. Here it is again. (The elephant does not appear on that graph, because its mass is too big.) The figure below shows the same function, graphed on log-log paper. Use this graph to estimate the mass and metabolic rate for the following animals, labeled on the graph.
Animal
Mouse
Dog
Sheep
Cow
Elephant
Mass (kg)
Metabolic rate (kcal/day)
Solution.
Animal
Mouse
Dog
Sheep
Cow
Elephant
Mass (kg)
\(0.02\)
\(15\)
\(50\)
\(500\)
\(4000\)
Metabolic rate (kcal/day)
\(3.5\)
\(500\)
\(1500\)
\(6000\)
\(50,000\)
Checkpoint5.102.Pause and Reflect.
If \(B=100A\text{,}\) the difference between \(A\) and \(B\) on a log scale is 2 units. Use the properties of logarithms to explain why this is true.
SubsectionAcidity and the pH Scale
You may have already encountered log scales in some everyday applications. A simple example is the pH scale, used by chemists to measure the acidity of a substance or chemical compound. This scale is based on the concentration of hydrogen ions in the substance, denoted by \([H^+]\text{.}\) The pH value is defined by the formula \begin{equation*} \text{pH}=-\log_{10}[H^+] \end{equation*} Values for pH fall between \(0\) and \(14\text{,}\) with \(7\) indicating a neutral solution. The lower the pH value, the more acidic the substance. Some common substances and their pH values are shown in the table.
Substance
pH
\([H^+]\)
Battery acid
\(1\)
\(0.1\)
Lemon juice
\(2\)
\(0.01\)
Vinegar
\(3\)
\(0.001\)
Milk
\(6.4\)
\(10^{-6.4}\)
Baking soda
\(8.4\)
\(10^{-8.4}\)
Milk of magnesia
\(10.5\)
\(10^{-10.5}\)
Lye
\(13\)
\(10^{-13}\)
Example5.103.
Calculate the pH of a solution with a hydrogen ion concentration of \(3.98 \times 10^{-5}\text{.}\)
The water in a swimming pool should be maintained at a pH of \(7.5\text{.}\) What is the hydrogen ion concentration of the water?
Solution.
We use a calculator to evaluate the pH formula with \([H^+] = 3.98\times10^{-5}\text{.}\) \begin{equation*} \text{pH} = -\log_{10}{(3.98 \times 10^{-5})} \approx 4.4 \end{equation*}
We solve the equation \begin{equation*} 7.5 = -\log_{10}[H^+] \end{equation*} for \([H^+]\text{.}\) First, we write \begin{equation*} -7.5 = \log_{10}[H^+] \end{equation*} Then we convert the equation to exponential form to get \begin{equation*} [H^+] = 10^{-7.5}\approx 3.2 \times 10^{-8} \end{equation*} The hydrogen ion concentration of the water is \(3.2 \times 10^{-8}\text{.}\)
Checkpoint5.104.Practice 5.
The pH of the water in a tide pool is \(8.3\text{.}\) What is the hydrogen ion concentration of the water? Solution.\(10^{-8.3}\approx 5.01\times 10^{-9}\)A decrease of \(1\) on the pH scale corresponds to an increase in acidity by a factor of \(10\text{.}\) Thus, lemon juice is \(10\) times more acidic than vinegar, and battery acid is \(100\) times more acidic than vinegar.
SubsectionDecibels
The decibel scale, used to measure the loudness or intensity of a sound, is another example of a logarithmic scale. The loudness of a sound is measured in decibels, \(D, \) by \begin{equation*} D=10 \log_{10}\left(\frac{I}{10^{-12}}\right) \end{equation*} where \(I\) is the intensity of its sound waves (in watts per square meter). The table below shows the intensity of some common sounds, measured in watts per square meter.
Sound
Intensity (watts/m\(^2\))
Decibels
Whisper
\(10^{-10}\)
\(20\)
Background music
\(10^{-8}\)
\(40\)
Loud conversation
\(10^{-6}\)
\(60\)
Heavy traffic
\(10^{-4}\)
\(80\)
Jet airplane
\(10^{-2}\)
\(100\)
Thunder
\(10^{-1}\)
\(110\)
Consider the ratio of the intensity of thunder to that of a whisper: \begin{equation*} \frac{\text{Intensity of thunder}}{\text{Intensity of a whisper}} = \frac{10^{-1}}{10^{-10}}= 10^9 \end{equation*} Thunder is \(10^9\text{,}\) or one billion times more intense than a whisper. It would be impossible to show such a wide range of values on a graph. When we use a log scale, however, there is a difference of only 90 decibels between a whisper and thunder.
Example5.105.
Normal breathing generates about \(10^{-11}\) watts per square meter at a distance of \(3\) feet. Find the number of decibels for a breath \(3\) feet away.
Normal conversation registers at about \(40\) decibels. How many times more intense than breathing is normal conversation?
Solution.
We evaluate the decibel formula with \(I = \alert{10^{-11}}\) to find \begin{equation*} \begin{aligned}[t] D \amp = 10 \log_{10}\left(\frac{\alert{10^{-11}}} {10^{-12}}\right) = 10 \log_{10} {10^1}\\ \amp = 10(1) = 10 \text{ decibels} \end{aligned} \end{equation*}
We let \(I_b\) stand for the sound intensity of breathing, and \(I_c\) stand for the intensity of normal conversation. We are looking for the ratio \(I_c/I_b\text{.}\) From part (a), we know that \begin{equation*} I_w = 10^{-11} \end{equation*} and from the formula for decibels, we have \begin{equation*} 40 = 10 \log_{10}\left(\frac{I_c}{10^{-12}}\right) \end{equation*} which we can solve for \(I_c\text{.}\) Dividing both sides of the equation by \(10\) and rewriting in exponential form, we have \begin{equation*} \begin{aligned}[t] \dfrac{I_c}{10^{-12}} \amp = 10^4\amp\amp \blert{\text{Multiply both sides by }10^{-12}.}\\ I_c \amp = 10^4(10^{-12}) = 10^{-8} \end{aligned} \end{equation*} Finally, we compute the ratio \(\dfrac{I_c}{I_b}\text{:}\) \begin{equation*} \frac{I_c}{I_b}= \frac{10^{-8}}{10^{-11}}= 10^3 \end{equation*} Normal conversation is \(1000\) times more intense than breathing.
Checkpoint5.106.Practice 6.
The noise of city traffic registers at about \(70\) decibels.
What is the intensity of traffic noise, in watts per square meter?
How many times more intense is traffic noise than conversation?
Solution.
\(I = 10^{-5}\) watts/m\(^2\)
\(\displaystyle 1000\)
Caution5.107.
Both the decibel model and the Richter scale in the next example use expressions of the form \(\log\left(\dfrac{a}{b}\right)\text{.}\) Be careful to follow the order of operations when using these models. We must compute the quotient \(\dfrac{a}{b}\) before taking a logarithm. In particular, recall that \(\log\left(\dfrac{a}{b}\right)\) is not equivalent to \(\dfrac{\log a}{\log b}\text{.}\)
SubsectionThe Richter Scale
One method for measuring the magnitude of an earthquake compares the amplitude \(A\) of its seismographic trace with the amplitude \(A_0\) of the smallest detectable earthquake. The log of their ratio is the Richter magnitude, \(M\text{.}\) Thus, \begin{equation*} M=\log_{10}\left(\frac{A}{A_0} \right) \end{equation*}
Example5.108.
The Northridge earthquake of January 1994 registered 6.9 on the Richter scale. What would be the magnitude of an earthquake 100 times as powerful as the Northridge quake?
How many times more powerful than the Northridge quake was the San Francisco earthquake of 1989, which registered 7.1 on the Richter scale?
Solution.
The amplitude \(A\) of the Northridge quake is given by \begin{equation*} 6.9 = \log_{10}\left(\dfrac{A}{A_0}\right) \end{equation*} and by rewriting in exponential form we find \begin{equation*} A = 10^{6.9}A_0 \end{equation*} An earthquake 100 times as powerful would have amplitude \begin{equation*} 100 A = 100 \cdot 10^{6.9}A_0 = 10^{8.9}A_0 \end{equation*} Thus, the magnitude of the more powerful quake is \begin{align*} M \amp = \log_{10}\left(\dfrac{10^{8.9}A_0}{A_0}\right)\\ \amp = \log_{10} 10^{8.9} = 8.9 \end{align*}
In part (a) we used the Richter formula to find that the amplitude of the Northridge quake was \begin{equation*} A = 10^{6.9}A_0 \end{equation*} Similarly, the amplitude of the San Francisco quake was \begin{equation*} A = 10^{7.1}A_0 \end{equation*} So the ratio of their amplitudes is \begin{equation*} \frac{10^{7.1}A_0}{10^{6.9}A_0}= 10^{0.2} \end{equation*} The San Francisco earthquake was \(10^{0.2}\text{,}\) or approximately 1.58 times as powerful as the Northridge quake.
Checkpoint5.109.Practice 7.
In October 2005, a magnitude 7.6 earthquake struck Pakistan. How much more powerful was this earthquake than the 1989 San Francisco earthquake of magnitude 7.1? Solution.\(10^{.5}\approx 3.16\)
Note5.110.
An earthquake \(100\text{,}\) or \(10^2\text{,}\) times as strong is only two units greater in magnitude on the Richter scale. In general, a difference of \(K\) units on the Richter scale (or any logarithmic scale) corresponds to a factor of \(10^K\) units in the intensity of the quake.
Checkpoint5.111.QuickCheck 4.
How much stronger is magnitude 4 earthquake than a magnitude 2 earthquake?
Twice as strong.
Four times as strong.
16 times as strong.
100 times as strong.
Example5.112.
On a log scale, the weights of two animals differ by \(1.6\) units. What is the ratio of their actual weights?
Solution.
A difference of \(1.6\) on a log scale corresponds to a factor of \(10^{1.6}\) in the actual weights. Thus, the heavier animal is \(10^{1.6}\text{,}\) or \(39.8\) times as heavy as the lighter animal.
Checkpoint5.113.Practice 8.
Two points, labeled \(A\) and \(B\text{,}\) differ by \(2.5\) units on a log scale. What is the ratio of their decimal values? Solution.\(10^{2.5}\approx 316.2\)
Checkpoint5.114.Pause and Reflect.
Explain what negative values on a log scale mean.
SubsectionSection Summary
SubsubsectionVocabulary
Look up the definitions of new terms in the Glossary.
Log scale
Log-log paper
SubsubsectionCONCEPTS
A log scale is useful for plotting values that vary greatly in magnitude. We plot the log of the variable instead of the variable itself.
A log scale is a multiplicative scale: Each increment of equal length on the scale indicates that the value is multiplied by an equal amount.
The pH value of a substance is defined by the formula \begin{equation*} \text{pH}=-\log_{10}[H^+] \end{equation*} where \([H^+]\) denotes the concentration of hydrogen ions in the substance.
The loudness of a sound is measured in decibels, \(D\text{,}\) by \begin{equation*} D=10 \log_{10}\left(\frac{I}{10^{-12}}\right) \end{equation*} where \(I\) is the intensity of its sound waves (in watts per square meter).
The Richter magnitude, \(M\text{,}\) of an earthquake is given by \begin{equation*} M=\log_{10}\left(\frac{A}{A_0} \right) \end{equation*} where \(A\) is the amplitude of its seismographic trace and \(A_0\) is the amplitude of the smallest detectable earthquake.
A difference of \(K\) units on a logarithmic scale corresponds to a factor of \(10^K\) units in the value of the variable.
SubsubsectionSTUDY QUESTIONS
What numbers are used to label the axis on a log scale?
What does it mean to say that a log scale is a multiplicative scale?
Delbert says that \(80\) decibels is twice as loud as \(40\) decibels. Is he correct? Why or why not?
Which is farther on a log scale, the distance between \(5\) and \(15\text{,}\) or the distance between \(0.5\) and \(1.5\text{?}\)
SubsubsectionSKILLS
Practice each skill in the Homework problems listed.
Plot values on a log scale: #1–4, 9 and 10
Read values from a log scale: #5–8, 11–14, 19 and 20
Compare values on a log scale: #15–18
Use log scales in applications: #21–40
ExercisesHomework 5.4
1.
The log scale is labeled with powers of \(10\text{.}\) Finish labeling the tick marks in the figure with their corresponding decimal values.
The log scale is labeled with integer values. Label the tick marks in the figure with the corresponding powers of \(10\text{.}\)
2.
The log scale is labeled with powers of \(10\text{.}\) Finish labeling the tick marks in the figure with their corresponding decimal values.
The log scale is labeled with integer values. Label the tick marks in the figure with the corresponding powers of \(10\text{.}\)
3.
Plot the values on a log scale.
\(x\)
\(0.075\)
\(1.3\)
\(4200\)
\(87,000\)
\(6.5\times 10^7 \)
4.
Plot the values on a log scale.
\(x\)
\(4\times 10^{-4} \)
\(0.008\)
\(0.9\)
\(27\)
\(90 \)
5.
Estimate the decimal value of each point on the log scale.
6.
Estimate the decimal value of each point on the log scale.
7.
The log scale shows various temperatures in Kelvins. Estimate the temperatures of the events indicated.
8.
The log scale shows the size of various objects, in meters. Estimate the sizes of the objects indicated.
9.
Plot the values of \([H^+]\) in the section "Acidity and the pH Scale" on a log scale.
10.
Plot the values of sound intensity in the section "Decibels" on a log scale.
11.
The magnitude of a star is a measure of its brightness. It is given by the formula \begin{equation*} m = 4.83 - 2.5 \log L \end{equation*} where \(L\) is the luminosity of the star, measured in solar units. Calculate the magnitude of the stars whose luminosities are given in the figure.
12.
Estimate the wavelength, in meters, of the types of electromagnetic radiation shown in the figure.
13.
The risk magnitude of an event is defined by \(R = 10+ \log p\text{,}\) where \(p\) is the probability of the event occurring. Calculate the probability of each event.
The sun will rise tomorrow, \(R = 10\text{.}\)
The next child born in Arizona will be a boy, \(R = 9.7\text{.}\)
A major hurricane will strike North Carolina this year, \(R = 9.1\text{.}\)
A 100-meter asteroid will collide with Earth this year, \(R = 8.0\text{.}\)
You will be involved in an automobile accident during a 10-mile trip, \(R = 5.9\text{.}\)
A comet will collide with Earth this year, \(R = 3.5\text{.}\)
You will die in an automobile accident on a 1000-mile trip, \(R=2.3\)
You will die in a plane crash on a 1000-mile trip, \(R = 0.9\text{.}\)
14.
Have you ever wondered why time seems to pass more quickly as we grow older? One theory suggests that the human mind judges the length of a long period of time by comparing it with its current age. For example, a year is \(20\%\) of a \(5\)-year-old’s lifetime, but only \(5\%\) of a \(20\)-year-old’s, so a year feels longer to a \(5\)-year-old. Thus, psychological time follows a log scale, like the one shown in the figure.
Label the tick marks with their base \(10\) logarithms, rounded to \(3\) decimal places. What do you notice about the values?
By computing their logs, locate \(18\) and \(22\) on the scale
Four years of college seems like a long time to an \(18\)-year-old. What length of time feels the same to a \(40\)-year-old?
How long will the rest of your life feel? Let \(A\) be your current age, and let \(L\) be the age to which you think you will live. Compute the difference of their logs. Now move backward on the log scale an equal distance from your current age. What is the age at that spot? Call that age \(B\text{.}\) The rest of your life will feel the same as your life from age \(B\) until now.
Compute \(B\) using a proportion instead of logs.
15.
What number is halfway between \(10^{1.5}\) and \(10^2\) on a log scale?
What number is halfway between \(20\) and \(30\) on a log scale?
16.
What number is halfway between \(10^{3.0}\) and \(10^{3.5}\) on a log scale?
What number is halfway between \(500\) and \(600\) on a log scale?
17.
The distances to two stars are separated by \(3.4\) units on a log scale. What is the ratio of their distances?
18.
The populations of two cities are separated by \(2.8\) units on a log scale. What is the ratio of their populations?
19.
The probability of discovering an oil field increases with its diameter, defined to be the square root of its area. Use the graph to estimate the diameter of the oil fields at the labeled points, and their probability of discovery. (Source: Deffeyes, 2001)
20.
The order of a stream is a measure of its size. Use the graph to estimate the drainage area, in square miles, for streams of orders \(1\) through \(4\text{.}\) (Source: Leopold, Wolman, and Miller)
Exercise Group.
In Problems 21–40, use the appropriate formulas for logarithmic models.
21.
The hydrogen ion concentration of vinegar is about \(6.3\times 10^{-4}\text{.}\) Calculate the pH of vinegar.
22.
The hydrogen ion concentration of spinach is about \(3.2\times 10^{-6}\text{.}\) Calculate the pH of spinach.
23.
The pH of lime juice is \(1.9\text{.}\) Calculate its hydrogen ion concentration.
24.
The pH of ammonia is \(9.8\text{.}\) Calculate its hydrogen ion concentration.
25.
A lawn mower generates a noise of intensity \(10^{-2}\) watts per square meter. Find the decibel level of the sound of a lawn mower.
26.
A jet airplane generates \(100\) watts per square meter at a distance of \(100\) feet. Find the decibel level for a jet airplane.
27.
The loudest sound emitted by any living source is made by the blue whale. Its whistles have been measured at \(188\) decibels and are detectable \(500\) miles away. Find the intensity of the blue whale’s whistle in watts per square meter.
28.
The loudest sound created in a laboratory registered at \(210\) decibels. The energy from such a sound is sufficient to bore holes in solid material. Find the intensity of a \(210\)-decibel sound.
29.
At a concert by The Who in 1976, the sound level \(50\) meters from the stage registered \(120\) decibels. How many times more intense was this than a \(90\)-decibel sound (the threshold of pain for the human ear)?
30.
The loudest scientifically measured shouting by a human being registered \(123.2\) decibels. How many times more intense was this than normal conversation at \(40\) decibels?
31.
The pH of normal rain is \(5.6\text{.}\) Some areas of Ontario have experienced acid rain with a pH of \(4.5\text{.}\) How many times more acidic is acid rain than normal rain?
32.
The pH of normal hair is about \(5\text{,}\) the average pH of shampoo is \(8\text{,}\) and \(4\) for conditioner. Compare the acidity of normal hair, shampoo, and conditioner.
33.
How much more acidic is milk than baking soda? (Refer to the table in this section.)
34.
Compare the acidity of lye and milk of magnesia. (Refer to the table in this section.)
35.
In 1964, an earthquake in Alaska measured \(8.4\) on the Richter scale. An earthquake measuring \(4.0\) is considered small and causes little damage. How many times stronger was the Alaska quake than one measuring \(4.0\text{?}\)
36.
On April 30, 1986, an earthquake in Mexico City measured \(7.0\) on the Richter scale. On September 21, a second earthquake, this one measuring \(8.1\text{,}\) hit Mexico City. How many times stronger was the September quake than the one in April?
37.
A small earthquake measured \(4.2\) on the Richter scale. What is the magnitude of an earthquake three times as strong?
38.
Earthquakes measuring \(3.0\) on the Richter scale often go unnoticed. What is the magnitude of a quake \(200\) times as strong as a \(3.0\) quake?
39.
The sound of rainfall registers at \(50\) decibels. What is the decibel level of a sound twice as loud?
40.
The magnitude, \(m\text{,}\) of a star is a function of its luminosity, \(L\text{,}\) given by \begin{equation*} m = 4.83 - 2.5 \log L \end{equation*} If one star is \(10\) times as luminous as another star, is the difference in their magnitudes? PrevTopNext PreTeXt logo
On the other hand, we can plot the mass of the cat if we set tick marks at intervals of \(1\) kg, but the axis will have to be extremely long to include even the wolf. We cannot show the masses of all these animals on the same scale 
To get around this problem, we’ll compute the the log of each mass, and use the logs on a new scale. The table below shows the base 10 log of each animal’s mass, rounded to \(2\) decimal places. 
We’d need to keep in mind that we are plotting the logs of the animals’ masses, and not the actual masses. However, remember that a logarithm is really an exponent! For example, the mass of the horse is 300 kg, and \begin{equation*} \text{since} ~~~\log_{10}(300) = {\blert{2.48}}, ~~~\text{then} ~~~ 10^{\blert{2.48}} = 300 \end{equation*} So instead of plotting the logs from the table, we will plot powers of 10 that give the actual masses of the animals, like this: 
Compare this new scale to the previous one. It looks almost the same, except that the number line is labeled with powers of 10. Even though we computed the log of each mass, we still plotted the actual mass of each animal, in its form as a power of 10. It is the scale on the number line that has changed. A scale labeled with powers of 10 is called a logarithmic scale, or log scale. The powers of 10 on a log scale are evenly spaced, so that the actual values at the tick marks look like this. 
We can see right away that the increments between tick marks on a log scale are not equal, as they are on a usual linear scale. The increments get larger as we move from left to right on the scale. However, when we are plotting powers of 10 we use the exponents to place the data points on the scale. For example, you can check that the mass of the horse, at \(10^{2.48} = 300\) kg, is plotted about half-way between \(10^2 = 100\) and \(10^3 = 1000\) on the log scale, because 2.48 is about half-way between 2 and 3. Similarly, the mass of the cat, at \(10^{0.60} = 4\) kg, is plotted between \(10^0 = 1\) and \(10^1 = 10\) on the log scale. 
Approximately how long ago did each of the following events occur? 
Once again, the difference between, say, \(10^{0.1}\) and \(10^{0.2}\) is not the same as the difference between \(10^{0.2}\) and \(10^{0.3}\text{.}\) The decimal values of the powers \(10^{0.1}\) through \(10^{0.9}\text{,}\) rounded to two places, are shown below. 
As we move from left to right on this scale, we multiply the value at the previous tick mark by \(10^{0.1}\text{,}\) or about \(1.258\text{.}\) For example, \begin{equation*} \begin{aligned}[t] 10^{0.2} \amp = 1.258 \times 10^{0.1} = 1.585\\ 10^{0.3} \amp = 1.585 \times 10^{0.1} = 1.995 \end{aligned} \end{equation*} and so on. Moving up by equal increments on a log scale does not add equal amounts to the values plotted; it multiplies the values by equal factors. 
Now, \(10^{1.5} = 10\sqrt{10}\text{,}\) or approximately \(31.62\text{.}\) Note how equal increments of \(0.5\) on the log scale correspond to equal factors of \(10^{0.5}\) in the values plotted: \begin{equation*} \begin{aligned}[t] 10 \times 3.162 \amp = 31.62~~~~ \text{ and }~~~~ 31.62 \times 3.162 = 100\\ 10^1 \times 10^{0.5} \amp = 101.5~~~~ \text{ and }~~~~ 101.5 \times 10^{0.5} = 10^2 \end{aligned} \end{equation*} 
On the log scale in Example 5.100, notice how the integer values are spaced: They get closer together as they approach the next power of \(10\text{.}\) You will often see log scales labeled not with powers of \(10\text{,}\) but with integer values, like this: 
In fact, log-log graph paper scales both axes with logarithmic scales. 
(The elephant does not appear on that graph, because its mass is too big.) The figure below shows the same function, graphed on log-log paper. 
Use this graph to estimate the mass and metabolic rate for the following animals, labeled on the graph. 
