Name ________________________________________ Date __________________ Class__________________
LESSON
4-5
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 ⎞
2 x + log ⎜
⎟=5
⎝ x ⎠
b. Simplify and solve the resulting equation.
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4-42
Holt McDougal Algebra 2
⎛ T ( t ) − TA ⎞
log ⎜
⎟
To − TA ⎠
⎝
=
= log b
t
Practice C
1. x = 2
2. x ≈ 1.414
3. x ≈ 3.4
4. x = 12
5. x = −3.75
6. x = −2
7. x = −0.5
8. x = −6.5
9. x = −0.5
10. x = 32.5
11. x = 20
12. x = 1
13. x = 8000
14. x = 109
15. x = 7 or x = −2
16. x = ±11
17. x = 5
19. x = ±20
1
2
21. x ≤ 1 or x ≥ 3
18. x = −0.1, 1
20. x = −3, or x =
= 10
⎛
⎛ T ( t ) − TA ⎞ ⎞
⎜ log ⎜⎜
⎟⎟ ⎟
⎜
⎝ To − TA ⎠ ⎟
⎜
⎟
t
⎜
⎟
⎜
⎟
⎝
⎠
=b
2. b ≈ 0.97854
3. First 3 steps same as #1;
⎛ T ( t ) − TA ⎞
⎜
⎟
To − TA ⎠
⎝
= t ; About 37 min
4. log
log b
5. b ≈ 0.98362
6. About 55 min
7. About 30 min
22. a. A = 9000(1.0425)t
Problem Solving
b. 20 years
1. a. 437 mg
c. $72,118.34
b. C(t) = C0(1 − 0.15)t
Reteach
c. 2.3 h
1. x = −2.5; 4−(−2.5) = 32
2. a. He can graph the equation 102 =
437(0.85)t and find the value of t
where C(t) is 102.
2. x ≈ 1.024; 34(1.024) ≈ 90.01
3. log 5x − 3 = log 600
b. y = 102 and y = 437(0.85)t
(x − 3) log 5 = log 600
c.
x ≈ 6.975
56.975 − 3 ≈ 600.352
4. 32 = x
x=9
5. 4x + 8 = 100
4x = 92
x = 23
⎛ 75 x ⎞
1
6. log ⎜
⎟ = 1; log 25x = 1; 10 = 25x;
3
⎝
⎠
2
10 = 25x; x =
5
3. D
Reading Strategies
Challenge
1. a. 24
1. T(t) − TA = [T0 − TA] bt
T ( t ) − TA
To − TA
4. H
b. 2x = 24; x = 4
c. Letting x = 4, 24 = 16
= bt
⎛ T ( t ) − TA ⎞
log ⎜⎜
⎟⎟ = t log b
⎝ To − TA ⎠
2. a. 102 = x
b. x = 100
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A46
Holt McDougal Algebra 2
c. log10100 = log10102 = 2
5
x
3. a. 3 =3
5
•
2 2
5
x
(3 ) = 3 = 3 • 3
3.
4
x+4
b. 3 = 3
c. 5 = x + 4; x = 1.
4. a.
4x + 2log 10x − 2log x = 10
Use of the Power
Property
2(2x + log 10x − log x) =
10
Factor left side
2x + log 10x − log x = 5
divide both sides
by 2
⎛ 10 x ⎞
2 x + log ⎜
⎟=5
⎝ x ⎠
use of the Quotient
Property
b. 2x + 1 = 5; x = 2
4-6 THE NATURAL BASE, E
5. x + 4
6. x
7. x3
8. (x + 1)5
9. x − 1
10. 3x
11. (5x)−1, or
12. 2x
13. $8080.37
1
5x
Practice B
Practice A
1.
1. a.
−2
−1
0
1
2
3
f(x) 7.4
2.7
1
0.37
0.14
0.05
x
4. 7x
b.
2.
2.
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A47
Holt McDougal Algebra 2