Download Problem Solving - Gull Lake Community Schools

Name _______________________________________ Date __________________ Class __________________
LESSON
7-5
Problem Solving
Exponential and Logarithmic Equations and Inequalities
While John and Cody play their favorite video game,
John drinks 4 cups of coffee and a cola, and Cody
drinks 2 cups of brewed tea and a cup of iced tea.
John recalls reading that up to 300 mg of caffeine is
considered a moderate level of consumption per day.
The rate at which caffeine is eliminated from the
bloodstream is about 15% per hour.
1. John wants to know how long it will take for the
caffeine in his bloodstream to drop to a moderate
level.
a. How much caffeine did John consume?
Caffeine Content of Some
Beverages
Beverage
Caffeine
(mg per
serving)
Brewed coffee
103
Brewed tea
36
Iced tea
30
Cola
25
________________________________________
b. Write an equation showing the amount of caffeine
in the bloodstream as a function of time.
c. How long, to the nearest tenth of an hour, will it
take for the caffeine in John’s system to reach
a moderate level?
2. a. Cody thinks that it will take at least 8 hours for the level of caffeine in
John’s system to drop to the same level of caffeine that Cody consumed.
Explain how he can use his graphing calculator to prove that.
________________________________________________________________________________________
b. What equations did Cody enter into
his calculator?
________________________________________
c. Sketch the resulting graph.
Choose the letter for the best answer.
3. About how long would it take for the
level of caffeine in Cody’s system to
drop by a factor of 2?
4. If John drank 6 cups of coffee and a cola,
about how long would it take for the level
of caffeine in his system to drop to a
moderate level?
A 0.2 hour
B 1.6 hours
F 0.5 hour
C 2.7 hours
G 1.6 hours
D 4.3 hours
H 4.7 hours
J 5.3 hour
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
7-41
Holt Algebra 2
Name _______________________________________ Date __________________ Class __________________
LESSON
7-5
Reading Strategy
Use Relationships
In solving equations with logarithms and exponents, first use the properties
of logarithms and exponential functions to simplify equations. Here are two
additional properties that are useful for solving equations.
• If x  y, then bx  by.
• If x  y, then logb x  logb y.
Use the equation 2x  16 for Exercise 1.
1. a. Express 16 as a power of 2.
b. Rewrite the equation so both sides have
the same base. What is the value of x?
c. Show how you can check your solution.
________________________________________________________________________________________
Use the equation log10 x  2 for Exercise 2.
2. a. Rewrite the equation using the definition of logarithm.
b. What is the solution of the equation?
c. What is the value of log 10 100?
Use the equation 243x  3x • 92 for Exercise 3.
3. a. Rewrite the equation so that the exponents on
both sides have the same base.
b. Simplify until it is in the form 3x  3y.
c. Solve for x.
Use the equation 4x  log (10x) 2  2log x  10 for Exercise 4.
4. a. Describe each step in the table to solve the equation.
4x  2log 10x  2log x  10
Use of the Power Property
2 (2x  log 10x  log x)  10
2x  log 10x  log x  5
 10 x 
2x  log 
5
 x 
b. Simplify and solve the resulting equation.
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
7-42
Holt Algebra 2
Name _______________________________________ Date __________________ Class __________________
LESSON
7-5
Reteach
Exponential and Logarithmic Equations and Inequalities
An exponential equation contains an expression that has a variable as an exponent.
5 x  25 is an exponential equation.
x  2, since 5 (2)  25.
Remember: You can take the logarithm of both sides of an exponential
equation. Then use other properties of logarithms to solve.
If x  y, then
log x  log y
(x  0 and y  0).
Solve 6x  2  500.
Step 1
Since the variable is in the exponent, take the log of both sides.
6x  2  500
log 6x  2  log 500
Step 2
Use the Power Property of Logarithms: log ap  p log a.
log 6x  2  log 500
(x  2) log 6  log 500
Step 3
“Bring down” the exponent to multiply.
Isolate the variable. Divide both sides by log 6.
(x  2) log 6  log 500
x2
Step 4
Solve for x. Subtract 2 from both sides.
x
Step 5
log 500
log 6
log 500
2
log 6
Use a calculator to approximate x.
x  1.468
Step 6
Use a calculator to check.
61.468  2  499.607
Solve and check.
1. 4x  32
2. 34x  90
3. 5x  3  600
log 4x  log 32
log 34x  log 90
x log 4  log 32
4x log 3  log 90
________________________
________________________
________________________
________________________
________________________
________________________
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
7-38
Holt Algebra 2
Name _______________________________________ Date __________________ Class __________________
LESSON
7-5
Reteach
Exponential and Logarithmic Equations and Inequalities (continued)
A logarithmic equation contains a logarithmic expression that has a variable.
log5 x  2 is a logarithmic equation.
x  25, since 52  25.
Combine and use properties of logarithms to solve logarithmic equations.
Solve: log 80x  log 4  1
Step 1
Use the Quotient Property of Logarithms.
log 80x  log 4  1
log
Step 2
logx  logy  log
80 x
1
4
x
y
Simplify.
log
80 x
1
4
log 20x  1
Step 3
Use the definition of the logarithm:
if bx  a, then logb a  x.
log10 20x  1
Remember: Use 10 as the base
when the base is not given.
101  20x
Step 4
Solve for x. Divide both sides by 20.
10  20x
1
x
2
Solve and check.
4. log3 x4  8
4 log3 x  8
log3 x 
8
4
5. log 4  log (x  2)  2
6. log 75x  log 3  1
log 4 (x  2)  2
log10 (4x  8)  2
4x  8  102
________________________
________________________
________________________
________________________
________________________
________________________
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
7-39
Holt Algebra 2
Practice B
LESSON
7-5
Exponential and Logarithmic Equations and Inequalities
Solve and check.
1. 52x  20
2. 122x  8  15
________________________
________________________
4. 16  64
x7
5x
3. 2x  6  4
0.2x
5. 243
________________________
 81
x5
________________________
x
 1 
8.  
 32 
 1
7.    162
2
________________________
2x
 64
________________________
________________________
6. 25x  125x – 2
________________________
 1 
9.  
 27 
x 6
 27
________________________
Solve.
10. log4 x5  20
11. log3 x6  12
________________________
13. log x  log 10  14
________________________
14. log x  log 5  2
________________________
16. log (x  4)  log 6  1
________________________
17. log x  log 25  2
2
________________________
________________________
12. log4 (x  6)3  6
________________________
15. log (x  9)  log (2x  7)
________________________
18. log (x  1)2  log (5x  1)
________________________
Use a table and graph to solve.
19. 2 x  5  64
________________________
20. log x3  12
________________________
21. 2x 3x  1296
________________________
Solve.
22. The population of a small farming community is declining at a rate of 7%
per year. The decline can be expressed by the exponential equation
P  C (1  0.07) t , where P is the population after t years and C is the current
population. If the population was 8,500 in 2004, when will the population be
less than 6,000?
Original content Copyright © by Holt McDougal. Additions and changes to the original content are the responsibility of the instructor.
7-39
Holt Algebra 2