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Lesson 5.3 Exercises, pages 364–368
A
3. Write each number as a power of 2.
a) 16
1
b) 128
ⴝ 24
c) 32
d) 1
ⴝ 2ⴚ5
ⴝ 27
ⴝ 20
4. Which numbers below can be written as powers of 5? Write each
number you identify as a power of 5.
a) 125
1
b) 10
ⴝ 53
c) 25
cannot be
written as a
power of 5
d) 1
ⴝ 5ⴚ2
ⴝ 50
5. Write each number as a power of 3. √
√
√
3
a) 3 9
b) 243
c) 3
1
ⴝ 93
1
ⴝ 32 # 3ⴚ1
1
ⴝ 32 ⴚ1
ⴝ 2432
1
ⴝ (32)3
ⴝ (35)2
5
2
ⴝ 32
ⴝ 33
√
d) 27 3
1
1
ⴝ 33 # 32
1
ⴝ 33 ⴙ2
1
7
1
ⴝ 32
ⴝ 3ⴚ2
B
6. Solve each equation.
a) 2x ⫽ 256
2x ⴝ 28
xⴝ8
34 ⴝ 3xⴙ1
4 ⴝ xⴙ1
xⴝ3
c) 3x ⫽ 9x⫺2
3x
x
x
x
ⴝ
ⴝ
ⴝ
ⴝ
(32)xⴚ2
2(x ⴚ 2)
2x ⴚ 4
4
e) 82x ⫽ 16x⫹3
(23)2x ⴝ (24)xⴙ3
3(2x)
6x
2x
x
ⴝ
ⴝ
ⴝ
ⴝ
b) 81 ⫽ 3x⫹1
4(x ⴙ 3)
4x ⴙ 12
12
6
©P DO NOT COPY.
d) 4x⫺1 ⫽ 2x⫹3
(22)xⴚ1 ⴝ 2xⴙ3
2(x ⴚ 1) ⴝ x ⴙ 3
2x ⴚ 2 ⴝ x ⴙ 3
x ⴝ5
f) 9x⫹1 ⫽ 243x⫹3
(32)xⴙ1 ⴝ (35)xⴙ 3
2(x ⴙ 1) ⴝ 5(x ⴙ 3)
2x ⴙ 2 ⴝ 5x ⴙ 15
ⴚ3x ⴝ 13
xⴝⴚ
13
3
5.3 Solving Exponential Equations—Solutions
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7. Use graphing technology to solve each equation. Give the solution
to the nearest tenth.
b) 3x ⫽ 100
a) 10 ⫽ 2x
Graph: y ⴝ 2x ⴚ 10
The approximate zero is
3.3219281
x ⬟ 3.3
c) 3x⫹1 ⫽ 50
Graph: y ⴝ 100 ⴚ 3x
The approximate zero is 4.1918065
x ⬟ 4.2
d) 30 ⫽ 2x⫺1
Graph: y ⴝ 50 ⴚ 3xⴙ1
The approximate zero is
2.5608768
x ⬟ 2.6
Graph: y ⴝ 2xⴚ1 ⴚ 30
The approximate zero is 5.9068906
x ⬟ 5.9
8. Explain why the equation 4x ⫽ ⫺2 does not have a real solution.
Verify, graphically, that there is no solution.
The value of a power with a positive base can never be negative, so the
equation does not have a real solution. When I graph y ⴝ ⴚ2 ⴚ 4x, the
graph does not have an x-intercept.
9. Solve each equation.
√
a) 2x = 8 3 2
1
1
34 # 32 ⴝ 3x
2x ⴝ 23 # 23
1
1
34 ⴙ2 ⴝ 3x
2x ⴝ 23ⴙ3
xⴝ3ⴙ
xⴝ
1
3
4ⴙ
10
3
√
c) 2x + 1 = 2 3 4
1
2
2x ⴙ 1 ⴝ 21 # 23
2
2x ⴙ 1 ⴝ 21ⴙ3
3
2
3
xⴝ
4
2x ⴝ
2
3
2
3
√
4
216 = 36x - 1
1
2164 ⴝ 62(x ⴚ 1)
1
(63)4 ⴝ 62(x ⴚ 1)
3
ⴝ 2x ⴚ 2
4
11
2x ⴝ
4
11
xⴝ
8
12
27
(32)x ⴝ (33) 2
2x ⴙ 1 ⴝ 2(22) 3
xⴝ
√
1
2x ⴙ 1 ⴝ 2 # 4
xⴙ1ⴝ1ⴙ
1
ⴝx
2
9
xⴝ
2
d) 9x =
1
3
e)
√
b) 81 3 = 3x
√
√
f) 1 72x + 1 = 3 49
1
1
72(x ⴙ 1) ⴝ (72)3
1
1
2
xⴙ ⴝ
2
2
3
3x ⴙ 3 ⴝ 4
3x ⴝ 1
xⴝ
1
3
5.3 Solving Exponential Equations—Solutions
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10. Solve each equation.
3
1
a) a4b = 2x
√
3
25
b) 5 = 25
x
1
5x ⴝ (52) 3 # 5 ⴚ 2
(2 ⴚ 2)3 ⴝ 2x
(ⴚ2)(3) ⴝ x
x ⴝ ⴚ6
√
3
49
c) 343 = 7x + 1
2
5x ⴝ 5 3 ⴚ2
2
ⴚ2
3
4
xⴝⴚ
3
xⴝ
x
√
1
d) a9b = 3 27
1
(72)3 # 7ⴚ3 ⴝ 7x ⴙ 1
2
e) 8
Quit
1
(3ⴚ2)x ⴝ 31 # (33)2
3
7 3 ⴚ3 ⴝ 7x ⴙ 1
3ⴚ2x ⴝ 31ⴙ 2
2
ⴚ3ⴝxⴙ1
3
7
ⴚ ⴝxⴙ1
3
10
xⴝⴚ
3
ⴚ2x ⴝ 1 ⴙ
1-x
√
3
16
= 4
ⴚ2x ⴝ
1
23(1 ⴚ x) ⴝ (24)3 # 2ⴚ2
4
23(1 ⴚ x) ⴝ 23 ⴚ 2
4
3 ⴚ 3x ⴝ ⴚ 2
3
11
ⴚ3x ⴝ ⴚ
3
11
xⴝ
9
5
2
xⴝⴚ
1
f) a8b
x+1
3
2
5
4
√
= 1 3 162x
(2ⴚ3)x ⴙ 1 ⴝ A(24)3B
1 x
4
3
ⴚ3x ⴚ 3 ⴝ x
ⴚ
13
xⴝ3
3
xⴝⴚ
9
13
11. Use graphing technology to solve each equation. Give the solution
to the nearest tenth.
a) 2 ⫽ 1.05x
Graph: y ⴝ 1.05x ⴚ 2
The approximate zero is
14.206699
x ⬟ 14.2
c) 2x⫹1 ⫽ 3x⫺2
Graph: y ⴝ 3xⴚ2 ⴚ 2xⴙ1
The approximate zero is
7.1285339
x ⬟ 7.1
©P DO NOT COPY.
x
b) 2 - 5 = 0.4
x
Graph: y ⴝ 0.4 ⴚ 2 ⴚ 5
The approximate zero is 6.6096405
x ⬟ 6.6
d) 3(2x) ⫽ 64
Graph: y ⴝ 64 ⴚ 3(2x)
The approximate zero is 4.4150375
x ⬟ 4.4
5.3 Solving Exponential Equations—Solutions
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12. A principal of $600 was invested in a term deposit that pays 5.5%
annual interest, compounded semi-annually. To the nearest tenth of
a year, when will the amount be $1000?
nt
Use: A ⴝ A0 a1 ⴙ nb
i
1000 ⴝ 600 a1 ⴙ
Substitute: A ⴝ 1000, A0 ⴝ 600, i ⴝ 0.055, n ⴝ 2
2t
0 .055
b
2
Graph y ⴝ 600a1 ⴙ
0.055
b
2
2t
ⴚ 1000, then determine the zero of the function.
The approximate zero is 9.4148676
It will take approximately 9.4 years for the term deposit to amount to $1000.
13. a) To the nearest year, how long will it take an investment of $500
to double at each annual interest rate, compounded annually?
i) 4%
ii) 6%
iv) 9%
v) 12%
nt
Use: A ⴝ A0 a1 ⴙ nb
i
iii) 8%
Substitute: A ⴝ 1000, A0 ⴝ 500, n ⴝ 1
1t
1000 ⴝ 500 a1 ⴙ b
i
1
2 ⴝ (1 ⴙ i)t
Use this expression below.
ii) Substitute: i ⴝ 0.06
i) Substitute: i ⴝ 0.04
t
2 ⴝ (1 ⴙ 0.06)t
2 ⴝ (1 ⴙ 0.04)
2 ⴝ 1.06t
2 ⴝ 1.04t
t
Graph: y ⴝ 1.06t ⴚ 2
Graph y ⴝ 1.04 ⴚ 2,
The approximate zero is:
then determine the zero of the
11.895661
function.
It will take approximately
The approximate zero is: 17.672988
12 years.
It will take approximately
18 years.
iii) Substitute: i ⴝ 0.08
2 ⴝ (1 ⴙ 0.08)t
2 ⴝ 1.08t
Graph: y ⴝ 1.08t ⴚ 2
The approximate zero is: 9.0064683
It will take approximately 9 years.
iv) Substitute: i ⴝ 0.09
2 ⴝ (1 ⴙ 0.09)t
2 ⴝ 1.09t
Graph: y ⴝ 1.09t ⴚ 2
The approximate zero is: 8.0432317
It will take approximately 8 years.
v) Substitute: i ⴝ 0.12
2 ⴝ (1 ⴙ 0.12)t
2 ⴝ 1.12t
Graph: y ⴝ 1.12t ⴚ 2
The approximate zero is: 6.1162554
It will take approximately 6 years.
b) What pattern is there in the interest rates and times in part a?
The product of each interest rate as a percent and time in years is 72.
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14. When light passes through glass, the intensity is reduced by 5%.
a) Determine a function that models the percent of light, P, that
passes through n layers of glass.
For 0 layers of glass, the percent of light is: P ⴝ 100
For 1 layer of glass, the percent of light is: P ⴝ 100(0.95)
For 2 layers of glass, the percent of light is: P ⴝ 100(0.95)2
For 3 layers of glass, the percent of light is: P ⴝ 100(0.95)3
For n layers of glass, the percent of light is: P ⴝ 100(0.95)n
b) Determine how many layers of glass are needed for only 25% of
light to pass through.
Solve the equation: 25 ⴝ 100(0.95)n
Graph a related function: y ⴝ 100(0.95)x ⴚ 25
The approximate zero of the function is: 27.026815
So, 27 layers of glass are needed.
C
15. Solve each equation, then verify the solution graphically.
a) 21x 2 ⫽ 16
b) 9x⫹4 ⫽ 31x 2
2
2
2(x ) ⴝ 24
x2 ⴝ 4
x ⴝ —2
The graph of y ⴝ 16 ⴚ 2(x )
has x-intercepts 2 and ⴚ2.
2
2
32(xⴙ4) ⴝ 3(x )
2x ⴙ 8 ⴝ x2
2
x ⴚ 2x ⴚ 8 ⴝ 0
(x ⴚ 4)(x ⴙ 2) ⴝ 0
x ⴝ 4 or x ⴝ ⴚ2
A graph of y ⴝ 3(x ) ⴚ 9xⴙ4 has
x-intercepts ⴚ2 and 4.
2
2
16. For what values of k does the equation 91x 2 ⫽ 27x⫹k have no real
2
solution?
9(x ) ⴝ 27xⴙk
3(2x ) ⴝ 33(xⴙk)
2x2 ⴝ 3x ⴙ 3k
2
2x ⴚ 3x ⴚ 3k ⴝ 0
For no real roots, the discriminant is less than 0.
(ⴚ3)2 ⴚ 4(2)(ⴚ3k)<0
(ⴚ3)2< 4(2)(ⴚ3k)
9<ⴚ24k
2
2
k<ⴚ
3
9
, or ⴚ
24
8
3
8
The equation has no real solution when k<ⴚ .
©P DO NOT COPY.
5.3 Solving Exponential Equations—Solutions
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