Apr 25, 1992 ... Notice that the graph does not cross the x-axis. ... Among the many applications
of exponential functions are the applications hav .... The following function gives
the cost of a can of Coca-Cola t years ..... 100 on a seismograph.
APPENDIX
B
EXPONENTIAL AND LOGARITHMIC FUNCTIONS Mathematics is the abstract key which turns the lock of the physical universe. John Polkinghorne
INTRODUCTION If you have had any problems with or had testing done on your thyroid gland, then you may have come in contact with radioactive iodine-131. Like all radioactive elements, iodine-131 decays naturally. The half-life of iodine-131 is 8 days, which means that every 8 days a sample of iodine-131 will decrease to half of its original amount. Table 1 and the graph in Figure 1 show what happens to a 1,600-microgram sample of iodine-131 over time.
A(t)
TABLE 1 Iodine-131 as a Function of Time A (Micrograms)
0
1,600
8
800
16
400
24
200
32
100
Micrograms
t (Days)
2,000 1,500 1,000 500 0
0
5
10 15 20 25 30 35
t
Days
Figure 1
The function represented by the information in the table and Figure 1 is A(t) ⫽ 1,600 ⭈ 2⫺t/8 It is one of the types of functions we will study in this appendix.
SECTION
B.1
EXPONENTIAL FUNCTIONS To obtain an intuitive idea of how exponential functions behave, we can consider the heights attained by a bouncing ball. When a ball used in the game of racquetball is 2 dropped from any height, the first bounce will reach a height that is ᎏ3ᎏ of the original 2 height. The second bounce will reach ᎏ3ᎏ of the height of the first bounce, and so on, as shown in Figure 1 on the following page.
464
Section B.1
Exponential Functions
465
h 2h 3
Bounce 1
2•2h 3 3
Bounce 2
Bounce Bounce 3 4
Figure 1 If the ball is dropped initially from a height of 1 meter, then during the first 2 bounce it will reach a height of ᎏ3ᎏ meter. The height of the second bounce will reach 2 2 ᎏᎏ of the height reached on the first bounce. The maximum height of any bounce is ᎏᎏ 3 3 of the height of the previous bounce. Initial height: Bounce 1: Bounce 2: Bounce 3: Bounce 4: . . . Bounce n:
h⫽1 2 2 h ⫽ ᎏ (1) ⫽ ᎏᎏ 3 3 2 h⫽ ᎏ 3 2 h⫽ ᎏ 3 2 h⫽ ᎏ 3
冢 冣 冢 冣 冢 冣 冢 冣 冢 冣 冢 冣
2 2 2 ᎏ ⫽ ᎏ 3 3 2 2 2 3 ᎏ ⫽ ᎏ 3 3 3 2 2 4 ᎏ ⫽ ᎏ 3 3 . . . 2 2 n⫺1 2 h⫽ ᎏ ᎏ ⫽ ᎏ 3 3 3
冢 冣
冢 冣
n
This last equation is exponential in form. We classify all exponential functions together with the following definition. DEFINITION
An exponential function is any function that can be written in the form f(x) ⫽ bx where b is a positive real number other than 1.
Each of the following is an exponential function: f(x) ⫽ 2x
y ⫽ 3x
冢 冣
1 f(x) ⫽ ᎏ 4
x
The first step in becoming familiar with exponential functions in general is to find some values for specific exponential functions.
466
Appendix B
Exponential and Logarithmic Functions
EXAMPLE 1
If the exponential functions f and g are defined by f(x) ⫽ 2x
g(x) ⫽ 3x
and
then f(0) ⫽ 20 ⫽ 1 f(1) ⫽ 21 ⫽ 2 f(2) ⫽ 22 ⫽ 4
g(0) ⫽ 30 ⫽ 1 g(1) ⫽ 31 ⫽ 3
1 f(⫺1) ⫽ 2⫺1 ⫽ ᎏ 2
g(2) ⫽ 32 ⫽ 9 g(3) ⫽ 33 ⫽ 27 1 g(⫺1) ⫽ 3⫺1 ⫽ ᎏ 3
1 1 f(⫺2) ⫽ 2⫺2 ⫽ ᎏ2 ⫽ ᎏ 2 4
1 1 g(⫺2) ⫽ 3⫺2 ⫽ ᎏ2 ⫽ ᎏ 3 9
1 1 f(⫺3) ⫽ 2⫺3 ⫽ ᎏ3 ⫽ ᎏ 2 8
1 1 g(⫺3) ⫽ 3⫺3 ⫽ ᎏ3 ⫽ ᎏ 3 27
f(3) ⫽ 23 ⫽ 8
In the introduction to this appendix we indicated that the half-life of iodine-131 is 8 days, which means that every 8 days a sample of iodine-131 will decrease to half of its original amount. If we start with A0 micrograms of iodine-131, then after t days the sample will contain A(t) ⫽ A0 ⭈ 2⫺t/8 micrograms of iodine-131.
EXAMPLE 2 A patient is administered a 1,200-microgram dose of iodine-131. How much iodine-131 will be in the patient’s system after 10 days and after 16 days? The initial amount of iodine-131 is A0 ⫽ 1,200, so the function that gives the amount left in the patient’s system after t days is
SOLUTION
A(t) ⫽ 1,200 ⭈ 2⫺t/8 After 10 days, the amount left in the patient’s system is A(10) ⫽ 1,200 ⭈ 2⫺10/8 ⫽ 1,200 ⭈ 2⫺1.25 ⬇ 504.5 mg After 16 days, the amount left in the patient’s system is A(16) ⫽ 1,200 ⭈ 2⫺16/8 ⫽ 1,200 ⭈ 2⫺2 ⫽ 300 mg We will now turn our attention to the graphs of exponential functions. Because the notation y is easier to use when graphing, and y ⫽ f(x), for convenience we will write the exponential functions as y ⫽ bx
Section B.1
EXAMPLE 3
Exponential Functions
467
Sketch the graph of the exponential function y ⫽ 2x.
Using the results of Example 1, we have the following table. Graphing the ordered pairs given in the table and connecting them with a smooth curve, we have the graph of y ⫽ 2x shown in Figure 2.
SOLUTION
x
y
⫺3
1 ᎏᎏ 8 1 ᎏᎏ 4 1 ᎏᎏ 2
⫺2 ⫺1 0 1 2 3
1 2 4 8
y 5 4 3 2 1
y = 2x
–5 –4 –3 –2 –1 –1 –2 –3 –4 –5
1 2 3 4 5
x
Figure 2 Notice that the graph does not cross the x-axis. It approaches the x-axis—in fact, we can get it as close to the x-axis as we want without it actually intersecting the x-axis. For the graph of y ⫽ 2x to intersect the x-axis, we would have to find a value of x that would make 2x ⫽ 0. Because no such value of x exists, the graph of y ⫽ 2x cannot intersect the x-axis. 1 x Sketch the graph of y ⫽ ᎏ . 3 SOLUTION The table shown here gives some ordered pairs that satisfy the equation. Using the ordered pairs from the table, we have the graph shown in Figure 3.
冢 冣
EXAMPLE 4
x ⫺3
y 27
⫺2 ⫺1 0 1
9 3 1
2 3
1 ᎏᎏ 3 1 ᎏᎏ 9 1 ᎏᎏ 27
y 5 4 3 2 1 –5 –4 –3 –2 –1 –1 –2 –3 –4 –5
( )x
y= 1 3
1 2 3 4 5
x
Figure 3 The graphs of all exponential functions have two things in common: (1) each crosses the y-axis at (0, 1) because b0 ⫽ 1; and (2) none can cross the x-axis because bx ⫽ 0 is impossible due to the restrictions on b.
468
Appendix B
Exponential and Logarithmic Functions
Figures 4 and 5 show some families of exponential curves to help you become more familiar with them on an intuitive level.
y
x y= 1 3 y x
()
y = 3x x y = 2x y=4
y= 1 2
()
5
5
4
4
3
3
2
2
1
1
–3 –2 –1
1
2
x
3
–3 –2 –1
( )x
y= 1 4
1
2
3
x
–1
–1
Figure 4
Figure 5
Among the many applications of exponential functions are the applications having to do with interest-bearing accounts. Here are the details.
COMPOUND INTEREST
If P dollars are deposited in an account with annual interest rate r, compounded n times per year, then the amount of money in the account after t years is given by the formula
冢
r A(t) ⫽ P 1 ⫹ ᎏ n
冣
nt
EXAMPLE 5 Suppose you deposit $500 in an account with an annual interest rate of 8% compounded quarterly. Find an equation that gives the amount of money in the account after t years. Then find the amount of money in the account after 5 years. First we note that P ⫽ 500 and r ⫽ 0.08. Interest that is compounded quarterly is compounded 4 times a year, giving us n ⫽ 4. Substituting these numbers into the preceding formula, we have our function
SOLUTION
冢
0.08 A(t) ⫽ 500 1 ⫹ ᎏ 4
冣
4t
⫽ 500(1.02)4t
To find the amount after 5 years, we let t ⫽ 5: A(5) ⫽ 500(1.02)4⭈5 ⫽ 500(1.02)20 ⬇ $742.97 Our answer is found on a calculator and then rounded to the nearest cent.
Section B.1
Exponential Functions
469
The Natural Exponential Function A commonly occurring exponential function is based on a special number we denote with the letter e. The number e is a number like p. It is irrational and occurs in many formulas that describe the world around us. Like p, it can be approximated with a decimal number. Whereas p is approximately 3.1416, e is approximately 2.7183. (If you have a calculator with a key labeled e x , you can use it to find e 1 to find a more accurate approximation to e.) We cannot give a more precise definition of the number e without using some of the topics taught in calculus. For the work we are going to do with the number e, we only need to know that it is an irrational number that is approximately 2.7183. Table 1 and Figure 6 show some values and the graph for the natural exponential function. y ⫽ f(x) ⫽ ex y
TABLE 1 x ⫺2 ⫺1 0 1 2 3
f (x) ⴝ ex 1 f (⫺2) ⫽ e ⫺2 ⫽ ᎏ2 ⬇ 0.135 e 1 f (⫺1) ⫽ e ⫺1 ⫽ ᎏ ⬇ 0.368 e f (0) ⫽ e 0 ⫽ 1 f (1) ⫽ e 1 ⫽ e ⬇ 2.72 f (2) ⫽ e 2 ⬇ 7.39 f (3) ⫽ e 3 ⬇ 20.1
5
y = ex
4 3 2 1 –3 –2 –1
1
2
3
x
–1
Figure 6 One common application of natural exponential functions is with interestbearing accounts. In Example 5 we worked with the formula
冢
r A⫽P 1⫹ ᎏ n
冣
nt
which gives the amount of money in an account if P dollars are deposited for t years at annual interest rate r, compounded n times per year. In Example 5 the number of compounding periods was 4. What would happen if we let the number of compounding periods become larger and larger, so that we compounded the interest every day, then every hour, then every second, and so on? If we take this as far as it can go, we end up compounding the interest every moment. When this happens, we have an account with interest that is compounded continuously, and the amount of money in such an account depends on the number e. CONTINUOUSLY COMPOUNDED INTEREST
If P dollars are deposited in an account with annual interest rate r, compounded continuously, then the amount of money in the account after t years is given by the formula A(t) ⫽ Pert
470
Appendix B
Exponential and Logarithmic Functions
EXAMPLE 6 Suppose you deposit $500 in an account with an annual interest rate of 8% compounded continuously. Find an equation that gives the amount of money in the account after t years. Then find the amount of money in the account after 5 years. The interest is compounded continuously, so we use the formula A(t) ⫽ Pert. Substituting P ⫽ 500 and r ⫽ 0.08 into this formula we have
SOLUTION
A(t) ⫽ 500e 0.08t After 5 years, this account will contain A(5) ⫽ 500e 0.08⭈5 ⫽ 500e 0.4 ⬇ $745.91 to the nearest cent. Compare this result with the answer to Example 5.
GETTING READY FOR CLASS
After reading through the preceding section, respond in your own words and in complete sentences. a. What is an exponential function? b. In an exponential function, explain why the base b cannot equal 1. (What kind of function would you get if the base was equal to 1?) c. Explain continuously compounded interest. 1 x
d. What characteristics do the graphs of y ⫽ 2x and y ⫽ 冢ᎏ2ᎏ冣 have in common?
PROBLEM SET B.1 1
Let f(x) ⫽ 3x and g(x) ⫽ (ᎏ2ᎏ)x, and evaluate each of the following: 1. g(0) 2. f(0) 3. g(⫺1) 4. g(⫺4) 5. f(⫺3) 6. f(⫺1) 7. f(2) ⫹ g(⫺2) 8. f (2) ⫺ g(⫺2) Graph each of the following functions. 9. y ⫽ 4x 10. ⫺x 11. y ⫽ 3 12. 13. y ⫽ 2x⫹1 14. 15. y ⫽ ex 16.
y ⫽ 2⫺x 1 ⫺x y ⫽ 冢ᎏ3ᎏ冣 y ⫽ 2x⫺3 y ⫽ e ⫺x
Graph each of the following functions on the same coordinate system for positive values of x only. 17. y ⫽ 2x, y ⫽ x 2, y ⫽ 2x 18. y ⫽ 3x, y ⫽ x 3, y ⫽ 3x Graph the family of curves y ⫽ bx using the given values of the base b. 1 1 1 1 20. b ⫽ ᎏ , ᎏ , ᎏ , ᎏ 2 4 6 8 21. Bouncing Ball Suppose the ball mentioned in the introduction to this section is dropped from a height of 6 feet above the ground. Find an exponential equation that gives the height h the ball will attain during the nth bounce (Figure 7). How high will it bounce on the fifth bounce?
19. b ⫽ 2, 4, 6, 8
Section B.1
Exponential Functions
471
h 2h 3
Bounce 1
2•2h 3 3
Bounce 2
Bounce Bounce 3 4
Figure 7 22. Bouncing Ball A golf ball is manufactured so that if it is dropped from A feet above the ground onto a hard surface, the maximum height of each bounce will 1 be ᎏ2ᎏ the height of the previous bounce. Find an exponential equation that gives the height h the ball will attain during the nth bounce. If the ball is dropped from 10 feet above the ground onto a hard surface, how high will it bounce on the 8th bounce? 23. Exponential Decay The half-life of iodine-131 is 8 days. If a patient is administered a 1,400-microgram dose of iodine-131, how much iodine-131 will be in the patient’s system after 8 days and after 11 days? (See Example 2.) 24. Exponential Growth Automobiles built before 1993 use Freon in their air conditioners. The federal government now prohibits the manufacture of Freon. Because the supply of Freon is decreasing, the price per pound is increasing exponentially. Current estimates put the formula for the price per pound of Freon at p(t) ⫽ 1.89(1.25)t, where t is the number of years since 1990. Find the price of Freon in 2000 and 2005. 25. Compound Interest Suppose you deposit $1,200 in an account with an annual interest rate of 6% compounded quarterly. a. Find an equation that gives the amount of money in the account after t years. b. Find the amount of money in the account after 8 years. c. If the interest were compounded continuously, how much money would the account contain after 8 years? 26. Compound Interest Suppose you deposit $500 in an account with an annual interest rate of 8% compounded monthly. a. Find an equation that gives the amount of money in the account after t years. b. Find the amount of money in the account after 5 years. c. If the interest were compounded continuously, how much money would the account contain after 5 years? 27. Compound Interest If $5,000 is placed in an account with an annual interest rate of 12% compounded four times a year, how much money will be in the account 10 years later? 28. Compound Interest If $200 is placed in an account with an annual interest rate of 8% compounded twice a year, how much money will be in the account 10 years later? 29. Bacteria Growth Suppose it takes 1 day for a certain strain of bacteria to reproduce by dividing in half. If there are 100 bacteria present to begin with, then the total number present after x days will be f(x) ⫽ 100 ⭈ 2x. Find the total number present after 1 day, 2 days, 3 days, and 4 days.
472
Appendix B
Exponential and Logarithmic Functions
30. Bacteria Growth Suppose it takes 12 hours for a certain strain of bacteria to reproduce by dividing in half. If there are 50 bacteria present to begin with, then the total number present after x days will be f(x) ⫽ 50 ⭈ 4x. Find the total number present after 1 day, 2 days, and 3 days. 31. Health Care In 1990, $699 billion were spent on health care expenditures. The amount of money, E, in billions spent on health care expenditures can be estimated using the function E(t) ⫽ 78.16(1.11)t, where t is time in years since 1970 (U.S. Census Bureau). a. How close was the estimate determined by the function to the actual amount of money spent on health care expenditures in 1990? b. What are the expected health care expenditures in 2005, 2006, and 2007? Round to the nearest billion. 32. Exponential Growth The cost of a can of Coca-Cola on January 1, 1960, was 10 cents. The following function gives the cost of a can of Coca-Cola t years after that. C(t) ⫽ 0.10e 0.0576t a. Use the function to fill in Table 2. (Round to the nearest cent.) TABLE 2 “Coca-Cola” is a registered trademark of The Coca-Cola Company. Used with the express permission of The CocaCola Company.
Years Since 1960 t 0 15 40 50
Cost C(t) $0.10
90
b. Use the table to find the cost of a can of Coca-Cola at the beginning of the year 2000. 33. Value of a Painting A painting is purchased as an investment for $150. If the painting’s value doubles every 3 years, then its value is given by the function V(t) ⫽ 150 ⭈ 2t/3
for t ⱖ 0
where t is the number of years since it was purchased, and V(t) is its value (in dollars) at that time. Graph this function. 34. Value of a Painting A painting is purchased as an investment for $125. If the painting’s value doubles every 5 years, then its value is given by the function V(t) ⫽ 125 ⭈ 2t/5
for t ⱖ 0
where t is the number of years since it was purchased, and V(t) is its value (in dollars) at that time. Graph this function. 35. Value of a Crane The function V(t) ⫽ 450,000(1 ⫺ 0.30)t where V is value and t is time in years, can be used to find the value of a crane for the first 6 years of use. a. What is the value of the crane after 3 years and 6 months? b. State the domain of this function. c. Sketch the graph of this function.
Section B.1
473
Exponential Functions
36. Value of a Printing Press The function V(t) ⫽ 375,000(1 ⫺ 0.25)t, where V is value and t is time in years, can be used to find the value of a printing press during the first 7 years of use. a. What is the value of the printing press after 4 years and 9 months? b. State the domain of this function. c. Sketch the graph of this function.
EXTENDING THE CONCEPTS
37. Drag Racing A dragster is equipped with a computer. The table gives the speed of the dragster every second during one race at the 1993 Winternationals. Figure 8 is a line graph constructed from the data in Table 3. 250
Elapsed Time (sec)
Speed (mi/hr)
TABLE 3 Speed of a Dragster Speed (mi/hr)
0
0.0
1
72.7
2
129.9
3
162.8
4
192.2
5
212.4
6
228.1
200 150 100 50 0
0
1
2 3 4 Time (sec)
5
6
Figure 8
The graph of the following function contains the first point and the last point shown in Figure 8; that is, both (0, 0) and (6, 228.1) satisfy the function. Graph the function to see how close it comes to the other points in Figure 8. s(t) ⫽ 250(1 ⫺ 1.5⫺t) 38. The graphs of two exponential functions are given in Figures 9 and 10. Use the graphs to find the following: a. f(0) b. f (⫺1) c. f (1) d. g(0) e. g(1) f. g(⫺1) y
y y = f(x)
y = g(x)
7 6
6
5
5
4
(–2, 4)
3
4 3
(1, 3)
2
(–1, 13 )
(–1, 2)
2
1 (0, 1)
–3 –2 –1
7
1
Figure 9
(0, 1)
(1, 12 )
1 2
3
x
–3 –2 –1
1
Figure 10
2
3
x
474
Appendix B
Exponential and Logarithmic Functions
SECTION
B.2
LOGARITHMS ARE EXPONENTS In January 1999, ABC News reported that an earthquake had occurred in Colombia, causing massive destruction. They reported the strength of the quake by indicating that it measured 6.0 on the Richter scale. For comparison, Table 1 gives the Richter magnitude of a number of other earthquakes. TABLE 1 Earthquakes Year
Earthquake
Richter Magnitude
1971
Los Angeles
6.6
1985
Mexico City
8.1
1989
San Francisco
7.1
1992
Kobe, Japan
7.2
1994
Northridge
6.6
1999
Armenia, Colombia
6.0
Although the sizes of the numbers in the table do not seem to be very different, the intensity of the earthquakes they measure can be very different. For example, the 1989 San Francisco earthquake was more than 10 times stronger than the 1999 earthquake in Colombia. The reason behind this is that the Richter scale is a logarithmic scale. In this section, we start our work with logarithms, which will give you an understanding of the Richter scale. Let’s begin. As you know from your work in the previous section, equations of the form y ⫽ bx
b ⬎ 0, b ⬆ 1
are called exponential functions. Every exponential function is a one-to-one function; therefore, it will have an inverse that is also a function. Because the equation of the inverse of a function can be obtained by exchanging x and y in the equation of the original function, the inverse of an exponential function must have the form x ⫽ by
b ⬎ 0, b ⬆ 1
Now, this last equation is actually the equation of a logarithmic function, as the following definition indicates: DEFINITION
The equation y ⫽ logb x is read “y is the logarithm to the base b of x” and is equivalent to the equation x ⫽ by
b ⬎ 0, b ⬆ 1
In words: y is the exponent to which b is raised in order to get x.
Section B.2
Logarithms Are Exponents
475
Notation When an equation is in the form x ⫽ by, it is said to be in exponential
form. On the other hand, if an equation is in the form y ⫽ logb x, it is said to be in logarithmic form. Table 2 shows some equivalent statements written in both forms. TABLE 2 Exponential Form 8⫽2 25 ⫽ 52
3
0.1 ⫽ 10⫺1 1 ᎏ ⫽ 2⫺3 8 r ⫽ zs
EXAMPLE 1 SOLUTION
Logarithmic Form ⇐⇒
log2 8 ⫽ 3
⇐⇒ ⇐⇒
log5 25 ⫽ 2 log10 0.1 ⫽ ⫺1 1 log2 ᎏ ⫽ ⫺3 8 logz r ⫽ s
⇐⇒ ⇐⇒
Solve log3 x ⫽ ⫺2.
In exponential form the equation looks like this: x ⫽ 3⫺2 1 x⫽ ᎏ 9
or 1 The solution is ᎏ 9.
EXAMPLE 2
Solve logx 4 ⫽ 3.
Again, we use the definition of logarithms to write the equation in exponential form: 4 ⫽ x3
SOLUTION
Taking the cube root of both sides, we have 3
3
兹苶4 ⫽ 兹x苶3 3
3
The solution is 兹4苶.
EXAMPLE 3 SOLUTION
x ⫽ 兹4苶 Solve log8 4 ⫽ x.
We write the equation again in exponential form: 4 ⫽ 8x
Because both 4 and 8 can be written as powers of 2, we write them in terms of powers of 2: 22 ⫽ (23)x 22 ⫽ 23x The only way the left and right sides of this last line can be equal is if the exponents are equal—that is, if 2 ⫽ 3x 2 or x⫽ ᎏ 3
476
Appendix B
Exponential and Logarithmic Functions
2
The solution is ᎏ3ᎏ. We check as follows: 2 log8 4 ⫽ ᎏ ⇐⇒ 4 ⫽ 82/3 3 3 4 ⫽ (兹 8苶)2 4 ⫽ 22 4⫽4
The solution checks when used in the original equation.
Graphing Logarithmic Functions Graphing logarithmic functions can be done using the graphs of exponential functions and the fact that the graphs of inverse functions have symmetry about the line y ⫽ x. Here’s an example to illustrate.
EXAMPLE 4
Graph the equation y ⫽ log2 x.
SOLUTION The equation y ⫽ log2 x is, by definition, equivalent to the exponen-
tial equation x ⫽ 2y which is the equation of the inverse of the function y ⫽ 2x We simply reflect the graph of y ⫽ 2x about the line y ⫽ x to get the graph of x ⫽ 2y, which is also the graph of y ⫽ log2 x. (See Figure 1.) y y = 2x
5 4 3 2 1 –5 –4 –3 –2 –1 –1 –2 –3 –4 –5
y=x
1 2 3 4 5
x
y = log2x or x = 2y
Figure 1 It is apparent from the graph that y ⫽ log2 x is a function because no vertical line will cross its graph in more than one place. The same is true for all logarithmic equations of the form y ⫽ logb x, where b is a positive number other than 1. Note also that the graph of y ⫽ logb x will always appear to the right of the y-axis, meaning that x will always be positive in the equation y ⫽ logb x.
Section B.2
Logarithms Are Exponents
477
USING TECHNOLOGY GRAPHING LOGARITHMIC FUNCTIONS
As demonstrated in Example 4, we can graph the logarithmic function y ⫽ log2 x as the inverse of the exponential function y ⫽ 2x. Your graphing calculator most likely has a command to do this. First, define the exponential function as Y1 ⫽ 2x. To see the line of symmetry, define a second function Y2 ⫽ x. Set the window variables so that ⫺6 ⱕ x ⱕ 6; ⫺6 ⱕ y ⱕ 6 Figure 2
and use your zoom-square command to graph both functions. Your graph should look similar to the one shown in Figure 2. Now use the appropriate command to graph the inverse of the exponential function defined as Y1 (Figure 3).
DrawInv Y1
Figure 3
Two Special Identities If b is a positive real number other than 1, then each of the following is a consequence of the definition of a logarithm: (1) b log b x ⫽ x
and
(2) log b bx ⫽ x
The justifications for these identities are similar. Let’s consider only the first one. Consider the equation y ⫽ logb x By definition, it is equivalent to x ⫽ by Substituting logb x for y in the last line gives us x ⫽ b log b x The next examples in this section show how these two special properties can be used to simplify expressions involving logarithms.
478
Appendix B
Exponential and Logarithmic Functions
EXAMPLE 5
Simplify 2log23.
SOLUTION Using the first property, 2log23 ⫽ 3.
EXAMPLE 6 SOLUTION
Simplify log10 10,000.
10,000 can be written as 104: log10 10,000 ⫽ log10 104 ⫽4
EXAMPLE 7 SOLUTION
Simplify logb b (b ⬎ 0, b ⬆ 1).
Because b 1 ⫽ b, we have logb b ⫽ logb b 1 ⫽1
EXAMPLE 8 SOLUTION
Simplify logb 1 (b ⬎ 0, b ⬆ 1).
Because 1 ⫽ b 0, we have logb 1 ⫽ logb b 0 ⫽0
EXAMPLE 9 SOLUTION
Simplify log4 (log5 5).
Because log5 5 ⫽ 1, log4 (log5 5) ⫽ log4 1 ⫽0
Application As we mentioned in the introduction to this section, one application of logarithms is in measuring the magnitude of an earthquake. If an earthquake has a shock wave T times greater than the smallest shock wave that can be measured on a seismograph (Figure 4), then the magnitude M of the earthquake, as measured on the Richter scale, is given by the formula M ⫽ log10 T (When we talk about the size of a shock wave, we are talking about its amplitude. The amplitude of a wave is half the difference between its highest point and its lowest point.) To illustrate the discussion, an earthquake that produces a shock wave that is 10,000 times greater than the smallest shock wave measurable on a seismograph will have a magnitude M on the Richter scale of M ⫽ log10 10,000 ⫽ 4
Section B.2
Sat Apr 25 1992 +90s
+180s
+270s +360s
+450s
Logarithms Are Exponents
+540s
+630s
+720s
+810s
479
+900s
GMT
14:00:00 16:00:00 18:00:00
BKS LHZ
20:00:00 22:00:00 00:00:00 02:00:00 04:00:00 06:00:00 08:00:00 10:00:00 12:00:00
Figure 4
EXAMPLE 10 If an earthquake has a magnitude of M ⫽ 5 on the Richter scale, what can you say about the size of its shock wave? SOLUTION
To answer this question, we put M ⫽ 5 into the formula M ⫽ log10 T
to obtain 5 ⫽ log10 T Writing this equation in exponential form, we have T ⫽ 105 ⫽ 100,000 We can say that an earthquake that measures 5 on the Richter scale has a shock wave 100,000 times greater than the smallest shock wave measurable on a seismograph. From Example 10 and the discussion that preceded it, we find that an earthquake of magnitude 5 has a shock wave that is 10 times greater than an earthquake of magnitude 4 because 100,000 is 10 times 10,000.
GETTING READY FOR CLASS
After reading through the preceding section, respond in your own words and in complete sentences. a. What is a logarithm? b. What is the relationship between y ⫽ 2x and y ⫽ log2 x? How are their graphs related? c. Will the graph of y ⫽ logb x ever appear in the second or third quadrants? Explain why or why not. d. Explain why log2 0 ⫽ x has no solution for x.
480
Appendix B
Exponential and Logarithmic Functions
PROBLEM SET B.2 Write each of the following equations in logarithmic form. 1. 24 ⫽ 16 2. 32 ⫽ 9 3. 125 ⫽ 53 1 5. 0.01 ⫽ 10⫺2 6. 0.001 ⫽ 10⫺3 7. 2⫺5 ⫽ ᎏ 32 1 ⫺3 1 ⫺2 3 9. ᎏ ⫽8 10. ᎏ ⫽9 11. 27 ⫽ 3 2 3
冢 冣
冢 冣
4. 16 ⫽ 42 1 8. 4⫺2 ⫽ ᎏ 16
12. 81 ⫽ 34
Write each of the following equations in exponential form. 13. log10 100 ⫽ 2 14. log2 8 ⫽ 3 15. log2 64 ⫽ 6 16. 18. log9 9 ⫽ 1 19. log6 36 ⫽ 2 20. 17. log8 1 ⫽ 0 21. log10 0.001 ⫽ ⫺3 22. log10 0.0001 ⫽ ⫺4 1 1 23. log5 ᎏ ⫽ ⫺2 24. log3 ᎏ ⫽ ⫺4 25 81 Solve each of the following equations for x. 25. log3 x ⫽ 2 26. log4 x ⫽ 3 27. log5 x ⫽ ⫺3 28. 29. log2 16 ⫽ x 30. log3 27 ⫽ x 31. log8 2 ⫽ x 32. 33. logx 4 ⫽ 2 34. logx 16 ⫽ 4 35. logx 5 ⫽ 3 36.
log2 32 ⫽ 5 log7 49 ⫽ 2
log2 x ⫽ ⫺4 log25 5 ⫽ x logx 8 ⫽ 2
Sketch the graph of each of the following logarithmic equations. 37. y ⫽ log3 x 38. y ⫽ log1/2 x 39. y ⫽ log1/3 x 40. y ⫽ log4 x Use your graphing calculator to graph each exponential function with the zoomsquare command. Then use the appropriate command to graph the inverse function, and write its equation in logarithmic form. 1 x 41. y ⫽ 5x 42. y ⫽ ᎏ 43. y ⫽ 10x 44. y ⫽ ex 5
冢 冣
Simplify each of the following. 45. log2 16 46. log3 9 49. log10 1,000 50. log10 10,000 53. log5 1 54. log10 1
47. log25 125 51. log3 3 55. log3 (log6 6)
48. log9 27 52. log4 4 56. log5 (log3 3)
Measuring Acidity In chemistry, the pH of a solution is defined in terms of logarithms as pH ⫽ ⫺log10 [H⫹] where [H⫹] is the hydrogen ion concentration in the solution. An acid solution has a pH below 7, and a basic solution has a pH higher than 7.
57. In distilled water, the hydrogen ion concentration is [H⫹] ⫽ 10⫺7. What is the pH? 58. Find the pH of a bottle of vinegar if the hydrogen ion concentration is [H⫹] ⫽ 10⫺3. 59. A hair conditioner has a pH of 6. Find the hydrogen ion concentration, [H⫹], in the conditioner. 60. If a glass of orange juice has a pH of 4, what is the hydrogen ion concentration, [H⫹], in the orange juice? 61. Magnitude of an Earthquake Find the magnitude M of an earthquake with a shock wave that measures T ⫽ 100 on a seismograph. 62. Magnitude of an Earthquake Find the magnitude M of an earthquake with a shock wave that measures T ⫽ 100,000 on a seismograph. 63. Shock Wave If an earthquake has a magnitude of 8 on the Richter scale, how many times greater is its shock wave than the smallest shock wave measurable on a seismograph?
Section B.2
Logarithms Are Exponents
481
64. Shock Wave If an earthquake has a magnitude of 6 on the Richter scale, how many times greater is its shock wave than the smallest shock wave measurable on a seismograph? EXTENDING THE CONCEPTS
65. The graph of the exponential function y ⫽ f(x) ⫽ bx is shown in Figure 5. Use the graph to complete parts a through d. y
64 48 32 16 –5 –4 –3 –2
(1, 8)
x
1 2 3 4 5
–16 –32 –48 –64
Figure 5 a. Fill in the table. x ⫺1 0 1
b. Fill in the table.
f (x)
x
f ⫺1(x) ⫺1 0 1
2
2
c. Find the equation for f(x). d. Find the equation for f ⫺1(x). 66. The graph of the exponential function y ⫽ f(x) ⫽ bx is shown in Figure 6. Use the graph to complete parts a through d. y
(–2, 25)
35 30 25 20 15 10
– 5 – 4 –3 –2 –1 –10 –15 –20 –25 –30 –35
1 2 3 4 5
Figure 6
x
482
Appendix B
Exponential and Logarithmic Functions
a. Fill in the table. x ⫺1 0 1
b. Fill in the table.
f (x)
x
f ⫺1(x) ⫺1 0 1
2
2
d. Find the equation for f ⫺1(x).
c. Find the equation for f(x).
SECTION
B.3
PROPERTIES OF LOGARITHMS If we search for the word decibel in Microsoft Bookshelf 98, we find the following definition: A unit used to express relative difference in power or intensity, usually between two acoustic or electric signals, equal to ten times the common logarithm of the ratio of the two levels. Figure 1 shows the decibel rating associated with a number of common sounds. Jet plane at takeoff Heavy traffic, thunder Rock music, subway Normal traffic, quiet train Noisy office Loud conversation Light traffic Normal conversation Quiet conversation Light whisper 0
10
20
30
40
50 60 Decibels
70
80
90
100
110
Figure 1 The precise definition for a decibel is I D ⫽ 10 log10 ᎏ I0
冢 冣
where I is the intensity of the sound being measured, and I0 is the intensity of the least audible sound. (Sound intensity is related to the amplitude of the sound wave that models the sound and is given in units of watts per meter2.) In this section, we will see that the preceding formula can also be written as D ⫽ 10(log10 I ⫺ log10 I0) The rules we use to rewrite expressions containing logarithms are called the properties of logarithms. There are three of them.
Section B.3
Properties of Logarithms
483
For the following three properties, x, y, and b are all positive real numbers, b ⫽ 1, and r is any real number.
PROPERTY 1
logb (xy) ⫽ logb x ⫹ logb y In words: The logarithm of a product is the sum of the logarithms.
PROPERTY 2
冢 冣
x logb ᎏ ⫽ logb x ⫺ logb y y
In words: The logarithm of a quotient is the difference of the logarithms.
PROPERTY 3
logb xr ⫽ r logb x In words: The logarithm of a number raised to a power is the product of the power and the logarithm of the number.
PROOF OF PROPERTY 1
To prove property 1, we simply apply the first identity for logarithms given in the preceding section: b logb xy ⫽ xy ⫽ (b logb x)(b logb y) ⫽ b logb x⫹logb y The first and last expressions are equal and the bases are the same, so the exponents logb xy and logb x ⫹ logb y must be equal. Therefore, logb xy ⫽ logb x ⫹ logb y The proofs of properties 2 and 3 proceed in much the same manner, so we will omit them here. The examples that follow show how the three properties can be used. 3xy Expand log5 ᎏ using the properties of logarithms. z SOLUTION Applying property 2, we can write the quotient of 3xy and z in terms of a difference:
EXAMPLE 1
3xy log5 ᎏ ⫽ log5 3xy ⫺ log5 z z
Applying property 1 to the product 3xy, we write it in terms of addition: 3xy log5 ᎏ ⫽ log5 3 ⫹ log5 x ⫹ log5 y ⫺ log5 z z
484
Appendix B
Exponential and Logarithmic Functions
EXAMPLE 2
Expand, using the properties of logarithms:
x4 log2 ᎏ3 兹y苶 ⭈ z 1/2 SOLUTION We write 兹y苶 as y and apply the properties: x4 x4 log2 ᎏ3 ⫽ log2 ᎏ 兹y苶 ⫽ y 1/2 y 1/2z 3 兹y苶 ⭈ z ⫽ log2 x 4 ⫺ log2 (y 1/2 ⭈ z 3) Property 2
⫽ log2 x 4 ⫺ (log2 y 1/2 ⫹ log2 z 3) Property 1 4 1/2 3 ⫽ log2 x ⫺ log2 y ⫺ log2 z Remove parentheses 1 ⫽ 4 log2 x ⫺ ᎏ log2 y ⫺ 3 log2 z Property 3 2 We can also use the three properties to write an expression in expanded form as just one logarithm.
EXAMPLE 3
Write as a single logarithm: 1 2 log10 a ⫹ 3 log10 b ⫺ ᎏ log10 c 3
SOLUTION
We begin by applying property 3:
1 2 log10 a ⫹ 3 log10 b ⫺ ᎏ log10 c 3
⫽ log10 a 2 ⫹ log10 b 3 ⫺ log10 c 1/3
Property 3
⫽ log10 (a ⭈ b ) ⫺ log10 c
Property 1
2
3
1/3
2 3
a b ⫽ log10 ᎏ c 1/3
Property 2
a 2b 3 ⫽ log10 ᎏ 3 兹 c苶
c 1/3 ⫽ 兹 c苶
3
The properties of logarithms along with the definition of logarithms are useful in solving equations that involve logarithms.
EXAMPLE 4
Solve log2 (x ⫹ 2) ⫹ log2 x ⫽ 3.
Applying property 1 to the left side of the equation allows us to write it as a single logarithm: log2 (x ⫹ 2) ⫹ log2 x ⫽ 3 log2 [(x ⫹ 2)(x)] ⫽ 3
SOLUTION
The last line can be written in exponential form using the definition of logarithms: (x ⫹ 2)(x) ⫽ 23 Solve as usual:
x⫹4⫽0 x ⫽ ⫺4
x 2 ⫹ 2x ⫽ 8 x 2 ⫹ 2x ⫺ 8 ⫽ 0 (x ⫹ 4)(x ⫺ 2) ⫽ 0 or x⫺2⫽0 or x⫽2
Section B.3
485
Properties of Logarithms
In the previous section we noted the fact that x in the expression y ⫽ logb x cannot be a negative number. Because substitution of x ⫽ ⫺4 into the original equation gives log2 (⫺2) ⫹ log2 (⫺4) ⫽ 3 which contains logarithms of negative numbers, we cannot use ⫺4 as a solution. The solution set is {2}.
GETTING READY FOR CLASS
After reading through the preceding section, respond in your own words and in complete sentences. a. Explain why the following statement is false: “The logarithm of a product is the product of the logarithms.” b. Explain why the following statement is false: “The logarithm of a quotient is the quotient of the logarithms.” c. Explain the difference between logb m ⫹ logb n and logb (m ⫹ n). Are they equivalent? d. Explain the difference between logb (mn) and (logb m)(logb n). Are they equivalent?
PROBLEM SET B.3 Use the three properties of logarithms given in this section to expand each expression as much as possible. 1. log3 4x
2. log2 5x
5. log2 y 5 9. log6 x 2y 4 xy 13. logb ᎏ z
6. log7 y 3 10. log10 x 2y 4 3x 14. logb ᎏ y
x 3兹y苶 17. log10 ᎏ z4
x 4兹 y苶 18. log10 ᎏ 兹z苶
5 3. log6 ᎏ x 3 7. log9 兹 z苶 11. log5 兹x苶 ⭈ y 4 4 15. log10 ᎏ xy
3
19. logb
Write each expression as a single logarithm. 21. logb x ⫹ logb z 22. 23. 2 log3 x ⫺ 3 log3 y 24. 1 1 25. ᎏ log10 x ⫹ ᎏ log10 y 26. 2 3
冪莦 3
x 2y ᎏ z4
x 4. log3 ᎏ 5 8. log8 兹z苶 3 12. log8 兹xy 苶6 5 16. log10 ᎏ 4y
20. logb
冪莦 4
x 4y 3 ᎏ z5
logb x ⫺ logb z 4 log2 x ⫹ 5 log2 y 1 1 ᎏ log10 x ⫺ ᎏ log10 y 3 4
1 27. 3 log2 x ⫹ ᎏ log2 y ⫺ log2 z 2 1 29. ᎏ log2 x ⫺ 3 log2 y ⫺ 4 log2 z 2
28. 2 log3 x ⫹ 3 log3 y ⫺ log3 z
3 3 4 31. ᎏ log10 x ⫺ ᎏ log10 y ⫺ ᎏ log10 z 2 4 5
4 32. 3 log10 x ⫺ ᎏ log10 y ⫺ 5 log10 z 3
30. 3 log10 x ⫺ log10 y ⫺ log10 z
486
Appendix B
Exponential and Logarithmic Functions
Solve each of the following equations. 33. log2 x ⫹ log2 3 ⫽ 1 34. log3 x ⫹ log3 3 ⫽ 1 35. log3 x ⫺ log3 2 ⫽ 2 36. log3 x ⫹ log3 2 ⫽ 2 37. log3 x ⫹ log3 (x ⫺ 2) ⫽ 1 38. log6 x ⫹ log6 (x ⫺ 1) ⫽ 1 39. log3 (x ⫹ 3) ⫺ log3 (x ⫺ 1) ⫽ 1 40. log4 (x ⫺ 2) ⫺ log4 (x ⫹ 1) ⫽ 1 41. log2 x ⫹ log2 (x ⫺ 2) ⫽ 3 42. log4 x ⫹ log4 (x ⫹ 6) ⫽ 2 2 2 43. log8 x ⫹ log8 (x ⫺ 3) ⫽ ᎏ 44. log27 x ⫹ log27 (x ⫹ 8) ⫽ ᎏ 3 3 苶 ⫹5⫽1 46. log2 兹x苶 ⫹ log2 兹6x 苶 ⫹5⫽1 45. log5 兹x苶 ⫹ log5 兹6x 47. Food Processing The formula M ⫽ 0.21(log10 a ⫺ log10 b) is used in the food processing industry to find the number of minutes M of heat processing a certain food should undergo at 250°F to reduce the probability of survival of Clostridium botulinum spores. The letter a represents the number of spores per can before heating, and b represents the number of spores per can after heating. Find M if a ⫽ 1 and b ⫽ 10⫺12. Then find M using the same values for a and b in the formula a M ⫽ 0.21 log10 ᎏ b P1 48. Acoustic Powers The formula N ⫽ log10 ᎏ is used in radio electronics to find P2 the ratio of the acoustic powers of two electric circuits in terms of their electric powers. Find N if P1 is 100 and P2 is 1. Then use the same two values of P1 and P2 to find N in the formula N ⫽ log10 P1 ⫺ log10 P2. 49. Henderson–Hasselbalch Formula Doctors use the Henderson–Hasselbalch formula to calculate the pH of a person’s blood. pH is a measure of the acidity and/or the alkalinity of a solution. This formula is represented as
冢 冣
x pH ⫽ 6.1 ⫹ log10 ᎏ y
where x is the base concentration and y is the acidic concentration. Rewrite the Henderson–Hasselbalch formula so that the logarithm of a quotient is not involved. 50. Henderson–Hasselbalch Formula Refer to the information in the preceding problem about the Henderson–Hasselbalch formula. If most people have a blood pH of 7.4, use the Henderson–Hasselbalch formula to find the ratio of x/y for an average person. 51. Decibel Formula Use the properties of logarithms to rewrite the decibel forI mula D ⫽ 10 log10 ᎏ so that the logarithm of a quotient is not involved. I0
冢 冣
I 52. Decibel Formula In the decibel formula D ⫽ 10 log10 ᎏ , the threshold of I0 hearing, I0, is
冢 冣
I0 ⫽ 10⫺12 watts/meter2 Substitute 10⫺12 for I0 in the decibel formula, and then show that it simplifies to D ⫽ 10(log10 I ⫹ 12)
Section B.4 Common Logarithms and Natural Logarithms
SECTION
B.4
487
COMMON LOGARITHMS AND NATURAL LOGARITHMS Acid rain was first discovered in the 1960s by Gene Likens and his research team, who studied the damage caused by acid rain to Hubbard Brook in New Hampshire. Acid rain is rain with a pH of 5.6 and below. As you will see as you work your way through this section, pH is defined in terms of common logarithms—one of the topics we present in this section. So, when you are finished with this section, you will have a more detailed knowledge of pH and acid rain.
HNO3 H2SO4 (acids)
SO2, NO2
Acid rain
Hundreds of miles
Common Logarithms Two kinds of logarithms occur more frequently than other logarithms. Logarithms with a base of 10 are very common because our number system is a base-10 number system. For this reason, we call base-10 logarithms common logarithms. DEFINITION
A common logarithm is a logarithm with a base of 10. Because common logarithms are used so frequently, it is customary, in order to save time, to omit notating the base; that is, log10 x ⫽ log x When the base is not shown, it is assumed to be 10. Common logarithms of powers of 10 are simple to evaluate. We need only recognize that log 10 ⫽ log10 10 ⫽ 1 and apply the third property of logarithms: logb xr ⫽ r logb x. log 1,000 ⫽ log 103 ⫽ 3 log 10 ⫽ 3(1) ⫽ 3 log 100 ⫽ log 102 ⫽ 2 log 10 ⫽ 2(1) ⫽ 2 log 10 ⫽ log 101 ⫽ 1 log 10 ⫽ 1(1) ⫽ 1 log 1 ⫽ log 100 ⫽ 0 log 10 ⫽ 0(1) ⫽ 0 log 0.1 ⫽ log 10⫺1 ⫽ ⫺1 log 10 ⫽ ⫺1(1) ⫽ ⫺1 log 0.01 ⫽ log 10⫺2 ⫽ ⫺2 log 10 ⫽ ⫺2(1) ⫽ ⫺2 log 0.001 ⫽ log 10⫺3 ⫽ ⫺3 log 10 ⫽ ⫺3(1) ⫽ ⫺3 To find common logarithms of numbers that are not powers of 10, we use a calculator with a log key.
488
Appendix B
Exponential and Logarithmic Functions
Check the following logarithms to be sure you know how to use your calculator. (These answers have been rounded to the nearest ten-thousandth.) log 7.02 ⬇ 0.8463 log 1.39 ⬇ 0.1430 log 6.00 ⬇ 0.7782 log 9.99 ⬇ 0.9996
EXAMPLE 1 SOLUTION
Use a calculator to find log 2,760.
log 2,760 ⬇ 3.4409
To work this problem on a scientific calculator, we simply enter the number 2,760 and press the key labeled log . On a graphing calculator we press the log key first, then 2,760. The 3 in the answer is called the characteristic, and the decimal part of the logarithm is called the mantissa.
EXAMPLE 2 SOLUTION
log 0.0391 ⬇ ⫺1.4078
EXAMPLE 3 SOLUTION
Find log 0.0391.
Find log 0.00523.
log 0.00523 ⬇ ⫺2.2815
EXAMPLE 4
Find x if log x ⫽ 3.8774.
We are looking for the number whose logarithm is 3.8774. On a scientific calculator, we enter 3.8774 and press the key labeled 10 x . On a graphing calculator we press 10 x first, then 3.8774. The result is 7,540 to four significant digits. Here’s why: If log x ⫽ 3.8774 then x ⫽ 103.8774
SOLUTION
⬇ 7,540 The number 7,540 is called the antilogarithm or just antilog of 3.8774. That is, 7,540 is the number whose logarithm is 3.8774.
EXAMPLE 5 SOLUTION
Find x if log x ⫽ ⫺2.4179.
Using the 10 x key, the result is 0.00382. If then
log x ⫽ ⫺2.4179 x ⫽ 10⫺2.4179 ⬇ 0.00382
The antilog of ⫺2.4179 is 0.00382; that is, the logarithm of 0.00382 is ⫺2.4179.
Applications Previously, we found that the magnitude M of an earthquake that produces a shock wave T times larger than the smallest shock wave that can be measured on a seismograph is
489
Section B.4 Common Logarithms and Natural Logarithms
given by the formula M ⫽ log10 T We can rewrite this formula using our shorthand notation for common logarithms as M ⫽ log T
EXAMPLE 6 The San Francisco earthquake of 1906 is estimated to have measured 8.3 on the Richter scale. The San Fernando earthquake of 1971 measured 6.6 on the Richter scale. Find T for each earthquake, and then give some indication of how much stronger the 1906 earthquake was than the 1971 earthquake. For the 1906 earthquake:
SOLUTION
If log T ⫽ 8.3, then T ⫽ 108.3 ⬇ 2.00 ⫻ 108 For the 1971 earthquake: If log T ⫽ 6.6, then T ⫽ 106.6 ⬇ 3.98 ⫻ 106
The shock wave for the 1906 earthquake was approximately 50 times stronger than the shock wave for the 1971 earthquake. In chemistry, the pH of a solution is the measure of the acidity of the solution. The definition for pH involves common logarithms. Here it is: pH ⫽ ⫺log [H⫹]
Figure 1
2
1
Most acidic rainfall recorded in U.S.
Battery acid
0
3 Apple juice Vinegar Lemon juice
4
5
Mean pH of Adirondack Lakes, 1975 Average pH of Killarney Lakes, 1971 Tomato juice Average pH of rainfall, Toronto, Feb. 1979
6
8
Increasingly acidic ACID RAIN
Blood NEUTRAL Milk Mean pH of Adirondack Lakes, 1930 Upper limit at which some fish affected "Clean" rain
Seawater, baking soda Lake Ontario
9
10
11
12 Ammonia
THE ACID SCALE
Milk of magnesia
13 Lye
Increasingly alkaline
7
NEUTRAL
where [H⫹] is the hydrogen ion concentration. The range for pH is from 0 to 14 (Figure 1). Pure water, a neutral solution, has a pH of 7. An acidic solution, such as vinegar, will have a pH less than 7, and an alkaline solution, such as ammonia, has a pH above 7.
14
8.3
Dividing the two values of T and rounding our answer to the nearest whole number, we have 2.00 ⫻ 108 ᎏᎏ6 ⬇ 50 3.98 ⫻ 10
490
Appendix B
Exponential and Logarithmic Functions
EXAMPLE 7 Normal rainwater has a pH of 5.6. What is the hydrogen ion concentration in normal rainwater? SOLUTION
Substituting 5.6 for pH in the formula pH ⫽ ⫺log [H⫹], we have
5.6 ⫽ ⫺log [H⫹]
Substitution
⫹
log [H ] ⫽ ⫺5.6
Isolate the logarithm
[H⫹] ⫽ 10⫺5.6
Write in exponential form
⬇ 2.5 ⫻ 10⫺6 moles per liter
Answer in scientific notation
EXAMPLE 8 The hydrogen ion concentration in a sample of acid rain known to kill fish is 3.2 ⫻ 10⫺5 moles per liter. Find the pH of this acid rain to the nearest tenth. SOLUTION
Substituting 3.2 ⫻ 10⫺5 for [H⫹] in the formula pH ⫽ ⫺log [H⫹],
we have pH ⫽ ⫺log [3.2 ⫻ 10⫺5]
Substitution
pH ⬇ ⫺(⫺4.5) ⫽ 4.5
Evaluate the logarithm Simplify
Natural Logarithms DEFINITION
A natural logarithm is a logarithm with a base of e. The natural logarithm of x is denoted by ln x; that is, ln x ⫽ loge x
The postage stamp shown in Figure 2 contains one of the two special identities we mentioned previously in this chapter, but stated in terms of natural logarithms.
Figure 2
Section B.4 Common Logarithms and Natural Logarithms
491
We can assume that all our properties of exponents and logarithms hold for expressions with a base of e because e is a real number. Here are some examples intended to make you more familiar with the number e and natural logarithms.
EXAMPLE 9
Simplify each of the following expressions.
a. e 0 ⫽ 1 b. e 1 ⫽ e c. ln e ⫽ 1 d. e. f. g.
In exponential form, e 1 ⫽ e
ln 1 ⫽ 0 In exponential form, e 0 ⫽ 1 ln e 3 ⫽ 3 ln e ⫺4 ⫽ ⫺4 ln et ⫽ t
EXAMPLE 10 sion ln Ae 5t. SOLUTION
Use the properties of logarithms to expand the expres-
The properties of logarithms hold for natural logarithms, so we have ln Ae 5t ⫽ ln A ⫹ ln e 5t ⫽ ln A ⫹ 5t ln e ⫽ ln A ⫹ 5t Because ln e ⫽ 1
EXAMPLE 11 a. ln 6
b. ln 0.5
If ln 2 ⫽ 0.6931 and ln 3 ⫽ 1.0986, find c. ln 8
SOLUTION
a. Because 6 ⫽ 2 ⭈ 3, using Property 1 for logarithms we have ln 6 ⫽ ln (2 ⭈ 3) ⫽ ln 2 ⫹ ln 3 ⫽ 0.6931 ⫹ 1.0986 ⫽ 1.7917 1
b. Writing 0.5 as ᎏ2ᎏ and applying property 2 for logarithms gives us 1 ln 0.5 ⫽ ln ᎏ 2 ⫽ ln 1 ⫺ ln 2 ⫽ 0 ⫺ 0.6931 ⫽ ⫺0.6931
c. Writing 8 as 23 and applying property 3 for logarithms, we have ln 8 ⫽ ln 23 ⫽ 3 ln 2 ⫽ 3(0.6931) ⫽ 2.0793
492
Appendix B
Exponential and Logarithmic Functions
GETTING READY FOR CLASS
After reading through the preceding section, respond in your own words and in complete sentences. a. What is a common logarithm? b. What is a natural logarithm? c. Is e a rational number? Explain. d. Find ln e, and explain how you arrived at your answer.
PROBLEM SET B.4 Find the following logarithms. 1. log 378 2. log 37.8 5. log 0.0378 6. log 37,800 9. log 0.00971 10. log 0.0314
3. log 3,780 7. log 600 11. log 0.00052
Find x in the following equations. 13. log x ⫽ 2.8802 14. log x ⫽ 4.8802 16. log x ⫽ ⫺3.1198 17. log x ⫽ ⫺5.3497 Find x without using a calculator. 19. log x ⫽ 10 20. log x ⫽ ⫺1 22. log x ⫽ 1 23. log x ⫽ 20 25. log x ⫽ ⫺2 26. log x ⫽ 4 28. log x ⫽ log3 9 Simplify each of the following expressions. 29. ln e 30. ln 1 31. ln e ⫺3
4. log 0.4260 8. log 10,200 12. log 0.399 15. log x ⫽ ⫺2.1198 18. log x ⫽ ⫺1.5670 21. log x ⫽ ⫺10 24. log x ⫽ ⫺20 27. log x ⫽ log2 8
32. ln ex
Use the properties of logarithms to expand each of the following expressions. 33. ln 10e 3t 34. ln 10e 4t 35. ln Ae ⫺2t 36. ln Ae ⫺3t Find a decimal approximation to each of the following natural logarithms. 37. ln 345 38. ln 3,450 39. ln 0.345 40. ln 0.0345 41. ln 10 42. ln 100 43. ln 45,000 44. ln 450,000 If ln 2 ⫽ 0.6931, ln 3 ⫽ 1.0986, and ln 5 ⫽ 1.6094, find each of the following. 1 1 45. ln 15 46. ln 10 47. ln ᎏ 48. ln ᎏ 3 5 49. ln 9 50. ln 25 51. ln 16 52. ln 81 Measuring Acidity Previously we indicated that the pH of a solution is defined in terms of logarithms as
pH ⫽ ⫺log [H⫹] where [H⫹] is the hydrogen ion concentration in that solution. 53. Find the pH of orange juice if the hydrogen ion concentration in the juice is [H⫹] ⫽ 6.50 ⫻ 10⫺4.
Section B.4 Common Logarithms and Natural Logarithms
493
54. Find the pH of milk if the hydrogen ion concentration in milk is [H⫹] ⫽ 1.88 ⫻ 10⫺6. 55. Find the hydrogen ion concentration in a glass of wine if the pH is 4.75. 56. Find the hydrogen ion concentration in a bottle of vinegar if the pH is 5.75. The Richter Scale Find the relative size T of the shock wave of earthquakes with the following magnitudes, as measured on the Richter scale.
57. 5.5 58. 6.6 59. 8.3 60. 8.7 61. Shock Wave How much larger is the shock wave of an earthquake that measures 6.5 on the Richter scale than one that measures 5.5 on the same scale? 62. Shock Wave How much larger is the shock wave of an earthquake that measures 8.5 on the Richter scale than one that measures 5.5 on the same scale? 63. Earthquake Table 1 gives a partial listing of earthquakes that were recorded in Canada in 2000. Complete the table by computing the magnitude on the Richter scale, M, or the number of times the associated shock wave is larger than the smallest measurable shock wave, T. TABLE 1 Location
Date
Moresby Island Vancouver Island Quebec City Mould Bay
January 23 April 30 June 29 November 13
St. Lawrence
December 14
Magnitude, M
Shock Wave, T
4.0 1.99 ⫻ 105 3.2 5.2 5.01 ⫻ 103
Source: National Resources Canada, National Earthquake Hazards Program.
64. Earthquake On January 6, 2001, an earthquake with a magnitude of 7.7 on the Richter scale hit southern India (National Earthquake Information Center). By what factor was this earthquake’s shock wave greater than the smallest measurable shock wave? Depreciation The annual rate of depreciation r on a car that is purchased for P dollars and is worth W dollars t years later can be found from the formula
W 1 log (1 ⫺ r) ⫽ ᎏ log ᎏᎏ P t 65. Find the annual rate of depreciation on a car that is purchased for $9,000 and sold 5 years later for $4,500. 66. Find the annual rate of depreciation on a car that is purchased for $9,000 and sold 4 years later for $3,000. Two cars depreciate in value according to the following depreciation tables. In each case, find the annual rate of depreciation. 67.
Age in Years new 5
Value in Dollars 7,550 5,750
68.
Age in Years new
Value in Dollars 7,550
3
5,750
494
Appendix B
Exponential and Logarithmic Functions
69. Getting Close to e Use a calculator to complete the following table. (1 ⫹ x)1/x
x 1 0.5 0.1 0.01 0.001 0.0001 0.00001
What number does the expression (1 ⫹ x)1/x seem to approach as x gets closer and closer to zero? 70. Getting Close to e Use a calculator to complete the following table. 1
冢1 ⫹ ᎏxᎏ冣
x
x
1 10 50 100 500 1,000 10,000 1,000,000
1 What number does the expression 1 ⫹ ᎏᎏ x and larger?
冢
SECTION
B.5
冣
x
seem to approach as x gets larger
EXPONENTIAL EQUATIONS AND CHANGE OF BASE For items involved in exponential growth, the time it takes for a quantity to double is called the doubling time. For example, if you invest $5,000 in an account that pays 5% annual interest, compounded quarterly, you may want to know how long it will take for your money to double in value. You can find this doubling time if you can solve the equation 10,000 ⫽ 5,000(1.0125)4t As you will see as you progress through this section, logarithms are the key to solving equations of this type. Logarithms are very important in solving equations in which the variable appears as an exponent. The equation 5x ⫽ 12
Section B.5
Exponential Equations and Change of Base
495
is an example of one such equation. Equations of this form are called exponential equations. Because the quantities 5x and 12 are equal, so are their common logarithms. We begin our solution by taking the logarithm of both sides: log 5x ⫽ log 12 We now apply property 3 for logarithms, log xr ⫽ r log x, to turn x from an exponent into a coefficient: x log 5 ⫽ log 12 Dividing both sides by log 5 gives us log 12 x ⫽ ᎏᎏ log 5 If we want a decimal approximation to the solution, we can find log 12 and log 5 on a calculator and divide: 1.0792 x ⬇ ᎏᎏ 0.6990 ⬇ 1.5439 The complete problem looks like this: 5x ⫽ 12 log 5x ⫽ log 12 x log 5 ⫽ log 12 log 12 x ⫽ ᎏᎏ log 5 1.0792 ⬇ ᎏᎏ 0.6990 ⬇ 1.5439 Here is another example of solving an exponential equation using logarithms.
EXAMPLE 1
Solve 252x⫹1 ⫽ 15.
SOLUTION Taking the logarithm of both sides and then writing the exponent
(2x ⫹ 1) as a coefficient, we proceed as follows: 252x⫹1 ⫽ 15 log 252x⫹1 ⫽ log 15
Take the log of both sides
(2x ⫹ 1) log 25 ⫽ log 15 log 15 2x ⫹ 1 ⫽ ᎏᎏ log 25 log 15 2x ⫽ ᎏᎏ ⫺ 1 log 25 1 log 15 x ⫽ ᎏᎏ ᎏᎏ ⫺ 1 2 log 25
冢
Property 3 Divide by log 25 Add ⫺1 to both sides
冣
1
Multiply both sides by ᎏ2ᎏ
496
Appendix B
Exponential and Logarithmic Functions
Using a calculator, we can write a decimal approximation to the answer: 1 1.1761 x ⬇ ᎏᎏ ᎏᎏ ⫺ 1 2 1.3979 1 ⬇ ᎏᎏ(0.8413 ⫺ 1) 2 1 ⫽ ᎏᎏ(⫺0.1587) 2 ⫽ ⫺0.0794
冢
冣
Recall from Section B.1 that if you invest P dollars in an account with an annual interest rate r that is compounded n times a year, then t years later the amount of money in that account will be r nt A ⫽ P 1 ⫹ ᎏᎏ n
冢
冣
EXAMPLE 2 How long does it take for $5,000 to double if it is deposited in an account that yields 5% interest compounded once a year? Substituting P ⫽ 5,000, r ⫽ 0.05, n ⫽ 1, and A ⫽ 10,000 into our formula, we have
SOLUTION
10,000 ⫽ 5,000(1 ⫹ 0.05)t 10,000 ⫽ 5,000(1.05)t 2 ⫽ (1.05)t
Divide by 5,000
This is an exponential equation. We solve by taking the logarithm of both sides: log 2 ⫽ log (1.05)t ⫽ t log 1.05 Dividing both sides by log 1.05, we have log 2 t ⫽ ᎏᎏ log 1.05 ⬇ 14.2 It takes a little over 14 years for $5,000 to double if it earns 5% interest per year, compounded once a year. There is a fourth property of logarithms we have not yet considered. This last property allows us to change from one base to another and is therefore called the change-of-base property.
PROPERTY 4 (CHANGE OF BASE)
If a and b are both positive numbers other than 1, and if x ⬎ 0, then logb x loga x ⫽ ᎏᎏ logb a ↑ ↑ Base a Base b
Section B.5
Exponential Equations and Change of Base
497
The logarithm on the left side has a base of a, and both logarithms on the right side have a base of b. This allows us to change from base a to any other base b that is a positive number other than 1. Here is a proof of property 4 for logarithms. PROOF
We begin by writing the identity a loga x ⫽ x Taking the logarithm base b of both sides and writing the exponent loga x as a coefficient, we have logb a loga x ⫽ logb x loga x logb a ⫽ logb x Dividing both sides by logb a, we have the desired result: loga x l ogb a log b x ᎏᎏ ⫽ ᎏᎏ l ogb a logb a log b x loga x ⫽ ᎏᎏ logb a We can use this property to find logarithms we could not otherwise compute on our calculators—that is, logarithms with bases other than 10 or e. The next example illustrates the use of this property.
EXAMPLE 3
Find log8 24.
We do not have base-8 logarithms on our calculators, so we can change this expression to an equivalent expression that contains only base-10 logarithms: log 24 log8 24 ⫽ ᎏᎏ Property 4 log 8
SOLUTION
Don’t be confused. We did not just drop the base, we changed to base 10. We could have written the last line like this: log10 24 log8 24 ⫽ ᎏᎏ log 10 8 From our calculators, we write 1.3802 log8 24 ⬇ ᎏᎏ 0.9031 ⬇ 1.5283
Application EXAMPLE 4 Suppose that the population in a small city is 32,000 in the beginning of 1994 and that the city council assumes that the population size t years later can be estimated by the equation P ⫽ 32,000e 0.05t Approximately when will the city have a population of 50,000?
498
Appendix B
Exponential and Logarithmic Functions
SOLUTION
We substitute 50,000 for P in the equation and solve for t: 50,000 ⫽ 32,000e 0.05t 1.5625 ⫽ e 0.05t
Divide both sides by 32,000
To solve this equation for t, we can take the natural logarithm of each side: ln 1.5625 ⫽ ln e 0.05t ⫽ 0.05t ln e ⫽ 0.05t ln 1.5625 t ⫽ ᎏᎏ 0.05 ⬇ 8.93 years
Property 3 for logarithms Because ln e ⫽ 1 Divide each side by 0.05
We can estimate that the population will reach 50,000 toward the end of 2002. USING TECHNOLOGY SOLVING EXPONENTIAL EQUATIONS
We can solve the equation 50,000 ⫽ 32,000e 0.05t from Example 4 with a graphing calculator by defining the expression on each side of the equation as a function. The solution to the equation will be the x-value of the point where the two graphs intersect. First define functions Y1 ⫽ 50000 and Y2 ⫽ 32000e (0.05x) as shown in Figure 1. Set your window variables so that 0 ⱕ x ⱕ 15; 0 ⱕ y ⱕ 70,000, scale ⫽ 10,000 Graph both functions, then use the appropriate command on your calculator to find the coordinates of the intersection point. From Figure 2 we see that the x-coordinate of this point is x ⬇ 8.93. 70,000 Plot1 Plot2 Plot3 \Y1=50000 \Y2=32000 e ˆ (0.05X) \Y3= \Y4= \Y5= \Y6= \Y7=
Intersection X=8.9257421 Y=50000
0
Figure 1
15
0
Figure 2
GETTING READY FOR CLASS
After reading through the preceding section, respond in your own words and in complete sentences. a. What is an exponential equation? b. How do logarithms help you solve exponential equations? c. What is the change-of-base property?
冢
冣
0.08 d. Write an application modeled by the equation A ⫽ 10,000 1 ⫹ ᎏᎏ 2
2⭈5
.
Section B.5
Exponential Equations and Change of Base
499
PROBLEM SET B.5 Solve each exponential equation. Use a calculator to write the answer in decimal form. 1. 3x ⫽ 5 2. 4x ⫽ 3 3. 5x ⫽ 3 4. 3x ⫽ 4 ⫺x ⫺x ⫺x 5. 5 ⫽ 12 6. 7 ⫽ 8 7. 12 ⫽ 5 8. 8⫺x ⫽ 7 x⫹1 x⫹1 x⫺1 9. 8 ⫽ 4 10. 9 ⫽ 3 11. 4 ⫽ 4 12. 3x⫺1 ⫽ 9 2x⫹1 2x⫹1 1⫺2x 13. 3 ⫽2 14. 2 ⫽3 15. 3 ⫽2 16. 21⫺2x ⫽ 3 3x⫺4 3x⫺4 5⫺2x 17. 15 ⫽ 10 18. 10 ⫽ 15 19. 6 ⫽4 20. 97⫺3x ⫽ 5 Use the change-of-base property and a calculator to find a decimal approximation to each of the following logarithms. 21. log8 16 22. log9 27 23. log16 8 24. log27 9 25. log7 15 26. log3 12 27. log15 7 28. log12 3 29. log8 240 30. log6 180 31. log4 321 32. log5 462 33. Compound Interest How long will it take for $500 to double if it is invested at 6% annual interest compounded twice a year? 34. Compound Interest How long will it take for $500 to double if it is invested at 6% annual interest compounded 12 times a year? 35. Compound Interest How long will it take for $1,000 to triple if it is invested at 12% annual interest compounded 6 times a year? 36. Compound Interest How long will it take for $1,000 to become $4,000 if it is invested at 12% annual interest compounded 6 times a year? 37. Doubling Time How long does it take for an amount of money P to double itself if it is invested at 8% interest compounded 4 times a year? 38. Tripling Time How long does it take for an amount of money P to triple itself if it is invested at 8% interest compounded 4 times a year? 39. Tripling Time If a $25 investment is worth $75 today, how long ago must that $25 have been invested at 6% interest compounded twice a year? 40. Doubling Time If a $25 investment is worth $50 today, how long ago must that $25 have been invested at 6% interest compounded twice a year? Recall that if P dollars are invested in an account with annual interest rate r, compounded continuously, then the amount of money in the account after t years is given by the formula A(t) ⫽ Pert. 41. Continuously Compounded Interest Repeat Problem 33 if the interest is compounded continuously. 42. Continuously Compounded Interest Repeat Problem 36 if the interest is compounded continuously. 43. Continuously Compounded Interest How long will it take $500 to triple if it is invested at 6% annual interest, compounded continuously? 44. Continuously Compounded Interest How long will it take $500 to triple if it is invested at 12% annual interest, compounded continuously? 45. Exponential Growth Suppose that the population in a small city is 32,000 at the beginning of 1994 and that the city council assumes that the population size t years later can be estimated by the equation P(t) ⫽ 32,000e 0.05t Approximately when will the city have a population of 64,000?
500
Appendix B
Exponential and Logarithmic Functions
46. Exponential Growth Suppose the population of a city is given by the equation P(t) ⫽ 100,000e 0.05t where t is the number of years from the present time. How large is the population now? (Now corresponds to a certain value of t. Once you realize what that value of t is, the problem becomes very simple.) 47. Exponential Growth Suppose the population of a city is given by the equation P(t) ⫽ 15,000e 0.04t where t is the number of years from the present time. How long will it take for the population to reach 45,000? Solve the problem using algebraic methods first, and then verify your solution with your graphing calculator using the intersection of graphs method. 48. Exponential Growth Suppose the population of a city is given by the equation P(t) ⫽ 15,000e 0.08t where t is the number of years from the present time. How long will it take for the population to reach 45,000? Solve the problem using algebraic methods first, and then verify your solution with your graphing calculator using the intersection of graphs method. EXTENDING THE CONCEPTS
49. 50. 51. 52. 53. 54.
Solve the formula A ⫽ Pert for t. Solve the formula A ⫽ Pe ⫺rt for t. Solve the formula A ⫽ P ⭈ 2⫺kt for t. Solve the formula A ⫽ P ⭈ 2kt for t. Solve the formula A ⫽ P(1 ⫺ r)t for t. Solve the formula A ⫽ P(1 ⫹ r)t for t.
