Name: __________________
Class:
x
1. Find the exponential function f (x) = a whose graph is given.
2. Find the exponential function f (x) = ax whose graph is given.
3. State the domain of the function f (x) = 5 x.
PAGE 1
Date: _____________
Name: __________________
Class:
Date: _____________
x
4. State the asymptote of the function g (x) = 9 7.
5. State the range of the function
h (x) = 8 +
1
7
x
6. Find the function of the form f (x) = Cax whose graph is given.
7. If $1000 is invested at an interest rate of 10% per year, compounded semiannually, find the value of the investment after 10 years.
8. The present value of a sum of money is the amount that must be invested now, at a given rate of interest, to produce the desired
sum at a later date. Find the present value of $10000 if interest is paid at a rate of 9% per year, compounded semiannually, for 9
years.
9. Find value of x at which the local minimum occurs for the function f (x) = e x + e
places.
PAGE 2
4 x
. State the answer correct to two decimal
Name: __________________
Class:
Date: _____________
10. Express the equation in exponential form
log 3 = 1
3
11. Express the equation ln (x + 5) = 2 in exponential form.
12. Express the equation in logarithmic form
10
2
= 100
13. Express the equation 5 z = n in logarithmic form.
14. Express the equation in logarithmic form
e
x + 2
= 0.9
15. Use the definition of the logarithmic function to find x:
log x = 0
6
16. Use the definition of the logarithmic function to find x:
log 9 = 2
x
PAGE 3
Name: __________________
Class:
Date: _____________
17. Find the function of the form y = log x whose graph is given.
a
PAGE 4
Name: __________________
Class:
Date: _____________
18. Find the function of the form y = log ax whose graph is given.
2
19. Find the domain of the function g (x) = log 3 (x 4).
4
20. Find the domain of the function f (x) = log (x x ) .
7
21. Find the domain of the function
f (x) =
PAGE 5
x 4 log ( 12 x )
5
Name: __________________
Class:
Date: _____________
22. Use the Laws of Logarithms to rewrite the expression
log ( x (x 9) )
3
in a form with no logarithm of a product.
23. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a quotient.
log
x
5
6
24. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a power.
log
10
13
9
25. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a power.
7
log
6
2
x +7
26. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a product, quotient or power.
log
x
a
8
yz
6
27. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a product or power.
6
ln
PAGE 6
5
3r s
Name: __________________
Class:
Date: _____________
28. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a product, quotient or power.
a
log
b
9
5
c
29. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a product, quotient or power.
ln
y
z
3
x
30. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a product, quotient or power.
x
log
(x
5
5
+ 9
+ 1)(x
3
2)
2
31. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a product or power.
5
x
log
5
y
5
z
32. Use the Laws of Logarithms to rewrite the expression below in a form with no logarithm of a product or power.
ln
z
9
y
2
5
x
+ 3y + 11
33. Evaluate the expression
log
PAGE 7
8
3
9
Name: __________________
Class:
Date: _____________
34. Rewrite the expression as a single logarithm
log 5 2 + 2 log 5 2
35. Rewrite the expression below as a single logarithm.
log 21 +
1
log 6 log 7
2
36. Rewrite the expression as a single logarithm
2
ln 6 + 3 ln x + 7 ln (x + 7)
37. Simplify (log 3 5)(log 5 19).
38. Find the solution of the exponential equation, correct to four decimal places.
x
e = 19
39. Find the solution of the exponential equation 4
4x 1
= 9, correct to four decimal places.
40. Find the solution of the exponential equation, correct to four decimal places.
7+3
41. Find the solution of the exponential equation e
2 2 x
= 14
= 11, correct to four decimal places.
42. Find the solution of the exponential equation 15 x = 2 x
PAGE 8
5x
+3
, correct to four decimal places.
Name: __________________
Class:
Date: _____________
43. Find the solution of the exponential equation below, correct to four decimal places.
11
x
= 2
3 + e
44. Find the solution of the exponential equation 1.00816
2x
= 10, correct to four decimal places.
45. Solve the equation
7
x 6x=6x
46. Solve the equation
2x
e 5 ex + 4 = 0
47. Solve the logarithmic equation for x:
ln x = 9
48. Solve the logarithmic equation for x:
log x = 1
49. Solve the logarithmic equation for x:
log ( 9 x + 7 ) = 2
50. Solve the logarithmic equation for x:
log 3( 4 x ) = 1
PAGE 9
Name: __________________
Class:
Date: _____________
51. Solve the logarithmic equation for x:
log 2 3 + log 2x = log 2 8 + log 2( x 25 )
52. Solve the logarithmic equation for x:
log 5( x + 7 ) log 5( x 7 ) = 2
53. For what value of x is the following true?
log ( x + 6 ) = log x + log 6
54. Solve for x.
log 2 ( log 3 x ) = 3
55. A sum of $10000 was invested for 7 years, and the interest was compounded semiannually. If this sum amounted to $12900 in
the given time, what was the interest rate?
56. A 25 g sample of radioactive iodine decays in such a way that the mass remaining after t days is given by
m ( t ) = 25 e
0.089 t
where m( t ) is measured in grams. After how many days is there only 8 g remaining?
PAGE 10
Name: __________________
Class:
Date: _____________
57. An electric circuit contains a battery that produces a voltage of 60 volts ( V ) , a resistor with a resistance of 13 ohms ( ) , and
an inductor with an inductance of 5 henrys ( H ), as shown in the figure. Using calculus, it can be shown that the current I = I ( t
60
13t/5
), ( in amps A ) t seconds after the switch is closed is I =
(1 e
) . Consider how you would express time as a
13
function of current, and apply that to answer the following question.
After how many seconds is the current 2.5 A? Enter the number of seconds rounded to three decimal places.
58. Use a graphing device to find all solutions of the equation, correct to three decimal places.
2
log x = x 2
59. Use a graphing device to find all solutions of the equation, correct to two decimal places.
e
x
2
2 = x
3
x
60. Solve the inequality.
log ( x 6 ) + log ( 13 x ) < 1
61. Solve the inequality.
2 < 10 x < 7
PAGE 11
Name: __________________
Class:
Date: _____________
62. Solve the inequality
2
x ex 25 ex < 0
63. Solve the equation
9x 3x
+1
=4
64. The fox population in a certain region has a relative growth rate of 8% per year. It is estimated that the population in 1998 was
11000. Find a function p(t) that models the population t years after 1998.
65. A culture starts with 8600 bacteria. After one hour the count is 11000. Find a function that models the number of bacteria n ( t )
after t hours.
66. An infectious strain of bacteria increases in number at a relative growth rate of 200% per hour. When a certain critical number of
bacteria are present in the bloodstream, a person becomes ill. If a single bacterium infects a person, the critical level is reached in
28 hours. How long (in hours) will it take for the critical level to be reached if the same person is infected with 18 bacteria?
67. The half life of cesium 137 is 30 years. Suppose we have a 85 g sample. Find a function that models the mass remaining after t
years.
68. Newton’s Law of Cooling is used in homicide investigations to determine the time of death. The normal body temperature is
o
98.6 F. Immediately following death, the body begins to cool. It has been determined experimentally that the constant in
Newton’s Law of Cooling is approximately k = 0.1947, assuming time is measured in hours. Suppose that the temperature of the
o
o
surroundings is 58 F. If the temperature of the body is now 74 F, how long ago ( in hours ) was the time of death? Round the
answer to the nearest tenth.
o
69. A kettle full of water is brought to a boil in a room with temperature 22 C . After 11 min the temperature of the water has
o
o
decreased from 100 C to 75 C . Find the temperature after another 11 min.
PAGE 12
Name: __________________
Class:
Date: _____________
70. An unknown substance has a hydrogen ion concentration of
+
[ H ] = 2.8 Find the pH.
PAGE 13
10
5
M
ANSWER KEY
Name: __________________
Class:
x
1. 4
x
1
3
2.
3. ( ,
4. y= 7
5. ( 8, )
6.
7.
8.
9.
)
x
4 3
$2653.29770514442230000000
$4528.00368847058000000000
0.28
1
10. 3 =3
2
11. x=e 5
12. log ( 100 ) =2
10
13. log ( n ) =z
5
14. x= 2+log ( 0.9 )
e
15. 1
16. x=3
17. y=log ( x )
2
18. y=log
19.
20.
21.
22.
3
2
( x)
( , 2 ) ( 2, )
( 0,1 )
4,12 )
log ( x ) +log ( x 9 )
3
3
23. log ( x ) log ( 5)
6
6
1
24.
log ( 13)
9
10
1
2
25. log x +7
6
7
26. 8log ( x ) log ( y ) 6log ( z )
a
a
a
(
27.
28.
29.
30.
31.
32.
33.
34.
)
1
5
1
ln ( 3) + ln ( r ) + ln ( s )
6
6
6
1
9log ( a ) 5log ( b) log ( c )
2
1
1
ln ( x ) + ln ( y ) ln ( z )
3
3
1
1
5
5
3
log x +9 log x +1 log x 2
10
10
10
2
2
1
1
1
log ( x ) + log ( y ) +
log ( z )
5
25
125
1
1
2
5ln ( z ) + ln ( x ) ln y +3y+11
2
9
1
4
log ( 8 )
(
)
(
(
5
35. log ( 3 6 )
( (
3
2
36. ln 6x x +7
PAGE 1
) 7)
)
(
)
)
Date: _____________
ANSWER KEY
Name: __________________
37. log ( 19 )
3
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.
53.
54.
55.
56.
57.
58.
59.
60.
61.
2.9444
0.6462
0.3542
0.1989
1.032
0.9163
141.6646
1
0,1.3863
8103.0839
0.1
10.33
1
40
7.58
1.2
6561
3.67103046082872800000%
13
0.300
0.01,1.472
0.89,0.71
x ( 6,8 ) ( 11,13)
log ( 2 ) ,log ( 7)
(
10
10
62. x ( 5,5)
63. 1.2619
0.08t
64. p=11000e
0.25t
65. n ( t ) =8600e
66. 26.55
67.
68.
69.
70.
0.023t
m ( t ) =85e
4.8
58
4.6
PAGE 2
)
Class:
Date: _____________