Name ___________________________________
Date _________________________
Additional Exercises 9.1
Form I
Exponential Functions
Approximate each number using a calculator. Round your answer to three decimal places.
1.
21.5
1. _______________
2.
4 2.3
2. _______________
3.
5 −1.2
3. _______________
Graph each function by making a table of coordinates.
4.
f ( x) = 2 x
5.
g ( x ) = −2 x
6.
&1#
h( x) = $ !
%3"
x
AE-333
Name ___________________________________
7.
f ( x) = 2 x − 3
8.
g ( x) = 4 x − 2
Date _________________________
Graph each function on the same rectangular coordinate system. Then describe how the graph
of g is related to the graph of f .
9.
f ( x) = 3 x
and
g ( x) = 3 x −1
9. _______________
10.
f ( x) = 3 x
and
g ( x ) = 3 x −3
10. _______________
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Name ___________________________________
11.
f ( x) = 3 x
and
Date _________________________
g ( x) = 3 x + 2
11. _______________
nt
& r#
Use the compound interest formulas A = P$1 + ! and A = Pe rt to solve.
% n"
12.
Find the accumulated value of an investment of $14,000
at 8% compounded annually for 11 years.
12. _______________
13.
Find the accumulated value of an investment of $14,000
at 8% compounded semiannually for 11 years.
13. _______________
14.
Find the accumulated value of an investment of $14,000
at 8% compounded continuously for 11 years.
14. _______________
15.
Find the accumulated value of an investment of $20,000
for 10 years at an interest rate of 4%, if the money is
compounded annually.
15. _______________
16.
Find the accumulated value of an investment of $17,000
for 2 years at an interest rate of 6%, if the money is
compounded quarterly.
16. _______________
17.
Find the accumulated value of an investment of $5,000
for 5 years at an interest rate of 4%, if the money is
compounded monthly.
17. _______________
18.
Find the accumulated value of an investment of $12,000
for 10 years at an interest rate of 3.5%, if the money is
compounded semiannually.
18. _______________
AE-335
Name ___________________________________
Date _________________________
Additional Exercises 9.2
Form I
Composite and Inverse Functions
Find the composition.
1.
If f ( x) = x 2 + 5 x and g ( x) = x + 2 , find ( f ! g )(3) .
1. ______________
2.
If f ( x) = 3 x − 4 and g ( x) = x + 5 , find ( g ! f )( x) .
2. ______________
3.
If f ( x) = 3 x 2 − 4 x and g ( x) = 2 x , find ( g ! f )( x) .
3. ______________
4.
If f ( x) = 2 x 2 + x and g ( x) = 3 x , find ( f ! g )( x) .
4. ______________
5.
If f ( x) =
x −8
and g ( x) = −3 x + 8 , find ( g ! f )( x) .
3
5. ______________
6.
If f ( x) =
x −8
and g ( x) = 3 x + 8 , find ( f ! g )( x) .
3
6. ______________
7.
If f ( x) =
x −8
and g ( x) = −3 x + 8 , find ( g ! f )(5) .
3
7. ______________
8.
If f ( x) = x + 6 and g ( x) = 8 x − 10 , find ( g ! f )( x) .
8. ______________
Determine whether the pair of functions f and g are inverses of each other.
9.
f ( x) = 6 x and g ( x) =
10.
f ( x) =
x
6
9. ______________
x+2
5
and g ( x) =
5
x+2
10. ______________
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Name ___________________________________
11.
f ( x) = x + 1 and g ( x) = x − 1
12.
f ( x) = 2 x − 5 and g ( x) =
13.
f ( x) =
14.
f ( x) = 3 x + 1 and g ( x) = −3 x − 1
Date _________________________
11. _______________
x+5
2
12. _______________
x−3
and g ( x) = 4 x + 3
4
13. _______________
14. _______________
Find the inverse of each one-to-one function. Write the inverse function using f
−1
( x ) notation.
15.
f ( x ) = −5 x − 7
15. _______________
16.
f ( x ) = −2 x
16. _______________
17.
f ( x) =
x−3
4
17. _______________
18.
f ( x) =
1
x +1
3
18. _______________
Graph each function and its inverse.
19.
f ( x ) = −3 x
20.
g ( x) =
1
x+2
2
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Name ___________________________________
Date _________________________
Additional Exercises 9.3
Form I
Logarithmic Functions
Write the equation in its equivalent exponential form.
1.
log 4 16 = 2
1. _______________
2.
log 2 8 = 3
2. _______________
3.
log 5 x = 2
3. _______________
4.
log 2 16 = x
4. _______________
Write the equation in its equivalent logarithmic form.
5.
32 = 9
5. _______________
6.
4 3 = 64
6. _______________
7.
52 = x
7. _______________
8.
6 y = 32
8. _______________
Evaluate the expression without using a calculator.
9.
log10 10
9. _______________
10.
log 7 1
10. _______________
11.
log10 1000
11. _______________
12.
log 36 6
12. _______________
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Name ___________________________________
13.
Date _________________________
13. _______________
log 2 2
Graph 14 and 15 on the same coordinate plane.
14.
f ( x) = 3 x
15.
g ( x) = log 3 x
16.
Find the domain of f ( x) = log 6 ( x + 5) .
16. _______________
Use inverse properties of logarithms to simplify each expression.
17.
ln e 2
17. _______________
18.
ln e15 x
18. _______________
19.
10 log
19. _______________
5x
Solve.
20.
&a#
Use the formula R = log$ ! + B to find the intensity R on
%T "
the Richter scale, given that amplitude a is 346 micrometers,
time T between waves is 2 seconds, and B is 2.2. Round your
answer to one decimal place.
AE-350
20. _______________
Name ___________________________________
Date _________________________
Additional Exercises 9.4
Form I
Properties of Logarithms
Use the properties of logarithms to expand the logarithmic expression as much as possible.
Where possible, evaluate logarithmic expressions without using a calculator.
1.
log 5 (7 ⋅ 11)
1. _______________
2.
log 2 3 x
2. _______________
3.
log 4 16 x
3. _______________
4.
log 7
5
x
4. _______________
5.
log 8
8
x
5. _______________
6.
log 5
125
x
6. _______________
7.
log 7 x 6
7. _______________
8.
ln(4 x)
8. _______________
9.
ln y 4
9. _______________
10.
log b
xy
z2
10. _______________
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Name ___________________________________
Date _________________________
Use the properties of logarithms to condense the logarithmic expression. Write the expression as
a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions.
11.
log 4 108 − log 4 9
11. _______________
12.
log 5 1250 − log 5 2
12. _______________
13.
log 6 2 + log 6 x
13. _______________
14.
log 6 9 + log 6 4
14. _______________
15.
log 7 ( x − 1) − log 7 ( x + 3)
15. _______________
16.
log b ( x + 2) − log b ( x − 2)
16. _______________
17.
2 log 5 x − 4 log 5 y
17. _______________
18.
6 log 3 x + 2 log 3 y
18. _______________
Use common logarithms or natural logarithms and a calculator to evaluate to four decimal
places.
19.
log 4 17
19. _______________
20.
log 6 14
20. _______________
AE-356
Name ___________________________________
Date _________________________
Additional Exercises 9.5
Form I
Exponential and Logarithmic Equations
Solve the exponential equation by expressing each side as a power of the same base and then
equating the exponents.
1.
5 x = 125
1. _______________
2.
4 x = 4096
2. _______________
3.
33 x −1 = 9
3. _______________
Solve each exponential equation by taking the logarithm on both sides. Express the solution set
in terms of logarithms.
4.
7 x = 343
4. _______________
5.
2 x = 90
5. _______________
6.
33 = 15
6. _______________
7.
e1.4 x = 10
7. _______________
8.
10 x = 9
8. _______________
Solve each exponential equation by taking the logarithm on both sides. Use a calculator to obtain
a decimal approximation correct to two decimal places for the solution.
9.
3 x+6 = 4
9. _______________
10.
e x = 72
10. _______________
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Name ___________________________________
Date _________________________
Solve each logarithmic equation. Be sure to reject any value that produces the logarithm of a
non-positive number in the original equation.
11.
log 2 ( x − 4) = −3
11. _______________
12.
log 4 (2 x + 6) = 3
12. _______________
13.
log 3 (2 x − 1) = 2
13. _______________
14.
log 6 x + log 6 ( x − 1) = 1
14. _______________
15.
log( x − 2) − log x = 3
15. _______________
Solve the equation by isolating the natural logarithm and exponentiating both sides. Express the
answer in terms of e.
16.
ln x = 2
16. _______________
17.
ln x = 24
17. _______________
18.
ln 2 x = 5
18. _______________
19.
6 + 5 ln x = 3
19. _______________
20.
6 ln 9 x = 12
20. _______________
AE-362
Name ___________________________________
Date _________________________
Additional Exercises 9.6
Form I
Exponential Growth and Decay: Modeling Data
Solve.
1. The value of a particular investment follows a pattern of exponential
growth. In the year 2008, you invested money in a money market
account. The value of your investment t years after 2008 is given by
the exponential growth model A = 2500e 0.051t . How much did you
initially invest in the account?
1. _______________
2. Using the exponential growth model in problem 1, A = 2500e 0.051t ,
how much will the account be worth in 2010? (Round to the nearest
cent.)
2. _______________
3. The value of a particular investment follows a pattern of exponential
growth. In the year 2005, you invested money in a money market
account. The value of your investment t years after 2005 is given by
the exponential growth model A = 4100e 0.066t . When will the account
be worth $5703?
3. _______________
4. The function A = A0 e −0.0099 x models the amount in pounds of a
4. _______________
particular radioactive material stored in a concrete vault, where A0
is the initial amount placed in the vault, and x is the number of years
since the material was put into the vault. If 400 pounds of the material
are initially put into the vault, how many pounds will be left after 100
years? (Round to the nearest pound.)
5. The function A = A0 e −0.00779 x models the amount in pounds of a
particular radioactive material stored in a concrete vault, where A0
is the amount of material initially placed in the vault, and x is the
number of years since the material was initially put into the vault. If
500 pounds of the material are initially put into the vault, how much
time will need to pass for only 116 pounds to remain?
5. _______________
6. The exponential growth model A = 8e 0.044t describes the population
of a particular country in millions, t years after 2000. What was the
population of the country in 2000?
6. _______________
7. Use the exponential growth model in problem 6, A = 8e 0.044t to find
the population of the country in 2010.
7. _______________
8. The half-life of silicon-32 is 710 years. If 20 grams is present now,
how much will be present in 900 years? (Round your answer to three
decimal places.)
8. _______________
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Name ___________________________________
Date _________________________
9. A fossilized leaf contains 36% of its normal amount of carbon 14.
How old is the fossil (to the nearest year)? Use 5600 years as the
half-life of carbon 14.
9. _______________
10. The half-life of radium is 1690 years. If 40 grams are present now,
how many grams will be present in 100 years?
10. _______________
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