A. 2 B. 1 C. 4 D. 3 - ARPDC Learning Portal

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A. 2 B. 1 C. 4 D. 3 - ARPDC Learning Portal
Math 30-1 Diploma Review
Exponents and Logarithms
1. The graph of f ( x)  b x , where b > 1, is translated such that the equation of the new graph is
expressed as y  2  f  x 1 . The range of the new function is
B. y  3
A. y  2
C. y  1
D. y  2
x
1
2. The graph of y  a x and the graph of y    , where a > 0, are reflections of each other
a
about the
A. x-axis
B. y-axis
C. line y = x
D. line y = -x
3. The y-intercept on the graph of f ( x)  a  x 1  b is
A. a
B. b
C. 1 + b
D. a + b
4. The graph of y  3x is reflected in the line y  x . The equation of the transformed graph is
A. y  3 x
B. y  3x
C. y  3log x
D. y  log3 x
5. The equation of the asymptote for the graph of y  logb  x  3  2 , where b  0 and b  1, is
A. x  2
B. x  2
C. x  3
D. x  3
6. The graph of y  log x is transformed into the graph of y  4  log  x  5 by a translation of
5 units
i and 4 units ii .
The statement above is completed by the information in row
Row
A.
B.
C.
D.
i
right
left
right
left
ii
up
up
down
down
7. The graph of the logarithmic function f ( x)  log3  x  2  intersects the x-axis at
A. 1
B. 2
C. log 3 2
8. For the graph of y  logb  3x  12 , where 0 < b < 1, the domain is
A. x  4
B. x  4
C. x  12
D. log 3 3
D. x  12
9. The range of f  x   log2 x is
A. x  0
C. f  x   0
B. x  R
D. f  x   R
10. If y  5x , x  R , then the inverse function equation can be expressed as
A. y  log 5 5
11. The expression
A.
2
B. y  log 5 x
C. y  log 5
D. y  log x
C. log3 6
D. log3 30
log 3 36
is equivalent to
log 3 6
B. 6
12. Numerical Response: The value of log 5 625  3log 7 49  log 2
1
 log a 1 is ___.
16
13. The equation m log p n  q can be written in exponential form as
A. p  nq
m
q
m
C. p q  mn
B. p  n
14. The equation y  43 x can also be written as
log 3 x
log 4 x
A. y 
B. y 
4
3
15. If f  x   3log x , and expression for f
A.
1
3log x
1
C. x 
log3 y
4
D. p qm  n
D. x 
log 4 y
3
x  is
1
1
B. log
3
x
x
D. 10 x3
C. 10 3
16. The equation of the asymptote of the graph of y  4log5  x  4  5 is
A. x  4
B. x  5
D. y  5
C. y  4
17. The expression  3log x  3log x  is equivalent to
A. 3log x
2
B. 9log x
2
C. 3
log x 
2
D. 9
log x 
2
2
Use the following information to answer the next question.
Numbered logarithmic expressions.
1.
log 4 62
2.
log6 36
3.
log3 10
4.
log5 20
18. Numerical Response: The value of the numbered logarithms above in order from least to
greatest is ___, ____, ___, ___.
19. Express 2 log a  log 3b  log c as a single logarithm.
 a 2c 
A. log 

 3b 
 ac 
C. 2 log  
 3b 
 a 2c 
B. log  3 
 b 
D. log  a 2  3b  c 
20. Written as a single logarithm, 2 log x 
 x2 
A. log  3

y z
 x2 y3 
C. log 

 z 
log z
 3log y is
2
 xy 
B. 3log  
 z 

D. log x 2  z  y 3



D.
x  4y
21. Numerical Response: Given that log3 a  6 and log3 b  5 , the value of log 3 9ab 2 is ___.
22. If log a 2  x and log a 3  y , then log a 24 in terms of x and y is
A. 2 x  3 y
B. 3x  y
C. 4x  y
23. The value of x in the equation log x 6  log x  x  1  1 is
A. 2
B. 3
C. 5
D. 3, 2
3
24. An expression equivalent to y  log5  x  1 is
x
B. y  log    1
5
log 5
D. y 
log  x  1
log x
1
log 5
log  x  1
A. y 
C. y 
25. If 3log x 
A. 2
log 5
1
, then the value of x is
9
1
B.
100
C. 2
D. 100
Use the following information to answer the next question.
A student’s work to simplify a logarithmic expression is shown below, where a > 1.
26. The student made his first error when going from:
A. Step 1 to Step 2
C. Step 3 to Step 4
B. Step 2 to Step 3
D. Step 4 to Step 5
27. Numerical Response: If b·c = a, then the value of log 𝑎 𝑏 + log 𝑎 𝑐 is ___.
28. If log2  3x  1  log2  x  3  log2 8 , then x is
A.
11
2
B. 5
C.
23
5
D. 3
4
29. If 63 x1  24 , then x can be expressed as
4 log 2  log 6
3 log 6
4 log 2  log 6
D.
3 log 6
3 log 6
4 log 2  log 6
3 log 6
C.
4 log 2  log 6
A.
B.
30. Determine the domain of the function y  log  2 x  3
A. x  
3
2
B. x  
2
3
C. x 
2
3
D. x 
3
2
31. If 0  a  1, which of the following is the best graph of y  log a x ?
32. If b·c = a, then the value of 2log 𝑎 𝑏 + 2 log 𝑎 𝑐 is
A. 2
B. 1
C. 4
D. 3
5
33. Which of the following expressions is equivalent to
 A2 B 

 C 
B. logC  A2 B 
A. logC  2 AB 
2log10 A  log10 B
log10 C
D. log10  2A  B  C 
C. log10 
34. Earthquake intensity is given by I  I o 10  where I o is the reference intensity and m is
magnitude. A recent earthquake in Washington measured a magnitude of 6.3 on the Richter
scale. In 1964, the Alaskan earthquake measured 8.5. How many times as intense was the
1964 Alaskan earthquake compared to the recent Washington earthquake?
m
A. 1.35
C. 101.35
B. 2.2
D. 10 2.2
35. The number of bacteria, n, present in a culture doubles every 5 days. Which equation could
be used to determine the number of bacteria present after t days if there are initially750
bacteria?
t
A. n  750  2 5
t
B. n  750  5 2
C. n  750  2 
D. n  750  5 
5t
2t
36. A culture starts with 3000 bacteria and after 3 h, it reaches 48 000 bacteria. For the culture,
the doubling time, to the nearest minute, is
A. 19 min
B.28 min
C.45 min
D.80 min
x
37. Numerical Response: The solution of 500  2.5  7.04 3 , to the nearest hundredth, is _____.
1
38. Numerical Response: The value of x in the equation  
8
hundredth, is ______.
x 3
 2 162 x 1  , to the nearest
39. Numerical Response: Earthquake intensity is given by I  I 0 10  , where I 0 is the
reference intensity and m is magnitude. A particular major earthquake of magnitude 7.9 is
120 times as intense as a particular minor earthquake. The magnitude, to the nearest tenth, of
the minor earthquake is ______
m
40. Numerical Response: The exact solution to the equation 8
a
simplified form as x   .
b
3 x  4
 4
x 9
can be expressed in
6
The value of a is ____
The value of b is ____
 2 x 1
41. Numerical Response: The solution to the equation 3
1
 
5
 x  3
can be expressed as
a log b  log 3
.
c log 3  log d
The value of a is ____
The value of b is ____
The value of c is ____
The value of d is ____
x
42. Which statement is true regarding the solutions to the equation log7  x  1  log7  x  5  1 ?
A The solutions are -2 and 6.
B. The solution is -2 since 6 is extraneous.
C. The solution is 6 since -2 is extraneous.
D. The solution is a null set.
Use the following information to answer the next question.
Earthquake intensity is given by I  I 0  10 M , where I 0 is the reference intensity and M is the
magnitude. An earthquake measuring 5.3 on the Richter scale is 125 times more intense than a
second earthquake.
43. Numerical Response: The Richter scale measure of the second earthquake, to the nearest
tenth, is ___.
44. Numerical Response: The population of a particular town was 20 000. If the population
decreases at an average annual rate of 1.4%, the number of entire years it take for the
population to reach 15 333 is ____.
45. If log 2 x  6 and log k x 
3
, then the value of k, to the nearest whole number, is ___.
2
Use the following information to answer the next question.
7
t
The estimated value of a particular painting is given by the equation V  V0  2.5 5 ,
where V0 is the original price, t is the number of years from the original purchase
date, and V is the value after t years.
46. Numerical Response: The number of years, to the nearest tenth, that it will take for the
painting to increase to 10 times its original value is ____ years.
Use the following information to answer the next question.
A student graphed the following equations:
Equation I
Equation II
Equation III
Equation IV
y  log10 x
y  5 x 3
y  x3
yx
47. A student could estimate the solution to the equation log5 x  x  3 by using the graphs of
equations
A. I and II
B. I and III
C. II and III
D. II and IV
48. The expression y  log a  a 4b   log a  ab  is equivalent to
A. 3
49. The equation log k B 
A.
m k
 B
n
B. 4
m
is equivalent to
n
m
B. B  k
n
C. 3a
D. a 3
C. B  n k m
D. k  n B m
50. If log3 y  c  log3 x , where y > 0 and x > 0, then y is equal to
A. c  y
B.
c
x
C.
c3
x
D.
3c
x
51. The graph of f  x   b x , b  0 is transformed to y  3 f  x  5  6 .The range of the
transformed graph is
A.  5,  
B.  , 5
C.  6,  
D.  , 6
8
52. The solution to the equation log x 6  log x  x  1  1 is
A.2
B.3
C.5
D. -3, 2
53. Greg invested $2000 at 4% per annum compounded monthly. How many years will it take
for the investment to grow to 14304?
A. 4 years
B. 5 years
C.6 years
D. 7 years
54. The half-life of a tracer element in the human body is 100 minutes. At noon, the mass of the
tracer element is 0.700 g. Correct to the nearest thousandth of a gram, the expected mass of
the same tracer element in 320 minutes is
A.0.093 g
B.0.076 g
C.0.066 g
D. 0.044 g
55. Given f ( x)  3log x , an expression for f 1  x  could be
A.
1
3log x
B.
1
1
log  
3
x
x
56. The graph of y  log5  x  1 is equivalent to the graph of
A. y 
log x
1
log 5
x
B. y  log    1
5
D. 10 x3
C. 10 3
C. y 
log  x  1
log 5
D.
log 5
log  x  1
57. The decibel level of a sound may be calculated using the formula L  10 log 1012 I  , where
L is the loudness of the sound (dB) and I is the intensity of the sound. An equation that could
be used to solve for the value of I is
L 120
L
L
L
12
A. I 
B. I  log    10
C. I  10 10
D. I  13
10
120 log10
 10 
58. The decibel level of a sound may be calculated using the formula L  10 log 1012 I  , where
L is the loudness of the sound (dB) and I is the intensity of the sound. The loudness of a jet
engine is 150 dB. The intensity is
A. 1.25
B. 1.18 1012
C. 1000
D. 1.5 1011
59. The pH of a solution can be calculated using the formula pH   log10 [ H  ] , where [ H  ] is
the concentration of hydronium ions in the solution in mol/L. The pH level of vinegar is 2.8.
The hydronium concentration of vinegar is 250 000 times greater than the hydronium
concentration in baking soda. An equation that could be used to determine the pH level of
baking soda, pH B is
A. 102.8 pH B  250000
C. 102.8 pH B  250000
B. 102.8 pH B  250000
D. 102.8 pH B  250000
9
Math 30-1 Diploma Review
Exponents and Logarithms Answers
1. A
2. B
3. D
4. D
5. C
6. B
7. A
8. A
9. D
10. B
11. A
12. 6
13. B
14. D
15. C
16. A
17. A
18. 4231
19. A
20. C
21. 18
22. B
23. A
24. C
25. B
26. B
27. 1
28. B
29. D
30. A
31. C
32. A
33. B
34. D
35. A
36. D
37. 8.14
38. 0.36
39. 5.8
40. 307
41. 3525
42. C
43. 3.2
44. 19
45. 16
46. 12.6
47. D
48. A
49. D
50. D
51. D
52. A
53. B
54. B
55. C
56. C
57. C
58. C
59. B

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