Download Exponential and Logarithmic Functions Diploma Review

Exponential and Logarithmic Functions: Math 30 – 1:
Diploma Review:
1. (4𝑥 −3 𝑦 5 )2 is equal to:
A.
B.
C.
D.
16𝑦 10
𝑥6
4𝑦 10
𝑥6
16𝑦 10
𝑥3
16𝑥 6
𝑦 10
1
2. (36𝑥 −4 )−2 is equal to:
A.
6
𝑥2
B. −18𝑥 2
C.
D.
𝑥2
6
𝑥 −4.5
6
3. When 16(8)𝑥−1 is converted to base 2, the exponent is:
A.
B.
C.
D.
3𝑥 + 1
3𝑥 + 3
7𝑥 − 7
12𝑥 − 12
1
4. If 42𝑥−7 = 64, then the value of √𝑥 is:
A. 2
B. √2
C. √5
D.
3
2
5. The solution to the equation 25𝑥+1 = 53(𝑥−1) , to the nearest tenth, is 𝑥 = ________.
1 𝑥−3
6. The solution to the equation (8)
= (2)(16)2𝑥+1, to the nearest hundredth, is
x = ______.
7. The solution to 82𝑥−1 = 16, to the nearest tent, is x =_____.
8. Which equation represents an exponential function?
A. 𝑦 = 2𝑥 8
B. 𝑦 = (−3)𝑥
C. 𝑦 =
3𝑥−2
2
1
D. 𝑦 = 3𝑥
Use the following to answer the next three questions:
A student is attempting to sketch the graph of the function 𝑓(𝑥) = 3𝑥−2 − 1 without
using a graphing calculator.
9. Which of the following is an asymptote of the graph?
A.
B.
C.
D.
𝑥 = −2
𝑥=2
𝑦=1
𝑦 = −1
10. The range of 𝑓 is:
A.
B.
C.
D.
𝑓(𝑥) ∈ 𝑅
{𝑓(𝑥)|𝑓(𝑥) > −2, 𝑓(𝑥) ∈ 𝑅}
{𝑓(𝑥)|𝑓(𝑥) ≥ −1, 𝑓(𝑥) ∈ 𝑅}
{𝑓(𝑥)|𝑓(𝑥) > −1, 𝑓(𝑥) ∈ 𝑅}
11. The x-intercept of the graph is:
A.
B.
C.
D.
0
2
3
There is no x-intercept
12. What is the y-intercept for the graph of y = bx - 2, b > 1?
A
.
1
b2
B.
-b2
C
.
1
b2
D.
2
13.In the equation y = bx, b > 1, x is replaced by x - 3 and y is replaced by y - 4. Which of the
following statements describes the transformation?
A. The point (x, y) on the graph of y = bx has been transformed to the point (x + 3, y + 4).
B. The point (x, y) on the graph of y = bx has been transformed to the point (x - 3, y - 4).
C. The graph of y = bx has been translated 4 units to the right and 3 units up.
D. The graph of y = bx has been translated 3 units to the left and 4 units down.
14. The graph of f(x) = ax, a > 1, is transformed into g(x) = 4ax + 3 - 2. Which characteristic
remains the same?
A. domain
B. range
C. x-intercept
D. y-intercept
15. The graph of the function f(x) = 3ax + 2, a > 0, has the same horizontal asymptote as
which of the following?
A. y = -f(x) - 4
B. y = -f(x) - 2
C. y = -f(x) + 2
D. y = -f(x) + 4
16. Mary was asked to solve for x and y in the exponential equations 5x + 3y = 1 and
1
25x  y  . Which of the following linear equations would lead to a correct solution?
5
A.
x + 3y = 1, x + y = -1
B.
x + 3y = 0, 2(x + y) = -1
C.
x + 3y = 1, 2x + y = -1
D.
x + 3y = 0, x + y = -2
17. Which function(s) would you graph to solve the equation 16
A. y1 = 16-0.5x, y2 = 0.54x + 3
B. y1  16
C. y1 


1
2
1
2x

4x  3

1
2
4x  3
1
 16 2
D. y1 = 4x, y2 

1
2
x
4x  3
1
 x
2


1
2
4x  3
graphically?
18. Given the function f(x) = 2x, match the graph with the correction equation.
Graph 1
Graph 2
Graph 3
Graph 4
Place the number of the graph that corresponds to y = -f(x) in the first box.
Place the number of the graph that corresponds to y = f(-x) in the first box.
Place the number of the graph that corresponds to y = f -1(x) in the first box.
Place the number of the graph that corresponds to y = -f(-x) in the first box.
19. The function f(x) = -5(2x) is transformed by a translation 2 units right and 5 units down. The
transformed function passes through the point (x, -10). Determine the value of x.
20. The graph of f (x) = logb x, b > 1, is translated such that the equation of the new graph is
expressed as y - 2 = f (x - 1). The domain of the new function is
A. {x | x > 0, x  R}
B. {x | x > 1, x  R}
C. {x | x > 2, x  R}
D. {x | x > 3, x  R}
21. The x-intercept of the function f (x) = log5 x + 3 is:
A. 5-3
B.
0
C. 1
D.
53
1
22. The equation y  log 2 x can also be written as:
3
x
y
A. y  2 3
B. x  2 3
C. 23x = y
D. 23y = x
23. The range of the inverse function, f -1, of f (x) = log4 x, is:
A. { y | y > 0, y  R}
B. { y | y < 0, y  R}
C. { y | y ≥ 0, y  R}
D. { y | y  R}
24. A graph of the function y = log3 x is transformed. The image of the point (3, 1) is (6, 3).
The equation of the transformed function is:
A. y = 3 log3 (x - 3)
B. y = 3 log3 (x + 3)
C. y - 3 = log3 (x - 3)
D. y + 3 = log3 (x + 3)
25. If log27 x = y, then log3 x equals:
A.
3y
B.
2
C. 3y
D.
2y
3
4y
26. If log 4 4096𝑥 = 64, then 𝑥 equals:
32
A.
B.
C.
D.
43
458
46
432
27. The graph of log 𝑥 𝑦 = 1 is:
28. To the nearest tenth, the y-intercept of the graph of log 5 (𝑦 + 2) = 𝑥 + 1 is _______.
2
29. If log 𝑏 81 = 3, then the value of 𝑏 to the nearest whole number is _____________.
30. Which of the following has a negative value?
A. − log 4 0.1
5
B. log 4 2
2
C. log 1 3
2
2
D. log 4 3
31. The value of the expression log √2 8 + 2log 9 3, to the nearest tenth, is ______.
32. Given the equation log 7 𝑥 = log 4 60, the value of x to the nearest whole number is ____.
33. If log 𝑥 27 = log 3 12, the value of x to the nearest whole number is
Use the following to answer the next question:
1
Three students were asked to find an alternate expression for log 𝑥 , 𝑥 > 0.
Alex gave the answer − log 𝑥
Bahman gave the answer log(−𝑥)
Connor gave the answer log(𝑥 −1 )
34. The correct alternative answer was given by:
A. Connor only
B. Alex and Connor
C. Baham and Connor
D. Some other combination of students
35. log 𝑥 + log(𝑥 + 4) is equal to:
A.
B.
C.
D.
log(2𝑥 + 4)
log(𝑥 2 + 4𝑥)
log(𝑥 2 + 4)
log(𝑥) log(𝑥 + 4)
36. log(𝑥 2 − 4) − log(𝑥 − 2) is equal to:
A. log(𝑥 + 2)
B. log(𝑥 2 − 𝑥 − 2)
C. log(𝑥 − 2)
D.
log(𝑥 2 −4)
log(𝑥−2)
37. (log 2𝑥)2 is equivalent to:
A.
B.
C.
D.
2 log 2𝑥
log 4𝑥 2
2 log 4𝑥
(log 2)2 + 2 log 2 log 𝑥 + (log 𝑥)2
1
38. The expression 3 log 𝑥 4 + log 𝑥 8 − 4 log 𝑥 16, where 𝑥 > 0. Is equal to:
A. log 𝑥 384
B.
3
4
log 𝑥 512
C. log 𝑥 256
D.
1
4
1
log 𝑥 (2)
39. log 𝑝 (𝑝6 𝑞 2 ) − log 𝑝 (𝑝2 𝑞 2 ) is equivalent to:
A.
B.
C.
D.
3
4
4p
p4
40. If log 3 𝐴 = 𝑡, then log 3 27𝐴3 =
A.
B.
C.
D.
3 + 3𝑡
3 + 𝑡3
9𝑡
3𝑡 3
1
41. If log 3 𝑥 2 = 2 and 2 log 𝑘 √𝑥 = 3, then the value of k is _____.
42. The graph of 𝑦 = log 𝑥 is transformed to the graph of 𝑦 = log(2𝑥 + 5) by a horizontal
stretch about the y-axis by a factor of p followed by a horizontal translation of q units
left. The value of p + q is:
A.
B.
C.
D.
3
4.5
5.5
7
43. An equation of the asymptote of 𝑦 = 3 log 4 (𝑥 + 2) − 6 is:
A.
B.
C.
D.
𝑥 = −6
𝑦 = −6
𝑥 = −2
𝑦 = −2
44. The graph of 𝑦 = log 7 𝑥 is reflected in the x-axis, the equation of the image can be
written in the form 𝑦 = log 𝑐 𝑥. The value of c, to the nearest hundredth, is ____.
45. A current, 𝐼𝑜 amperes, falls to 𝐼 amperes after t seconds according to the formula 𝐼 =
𝐼𝑜 𝑒 −𝑘𝑡 . The value of the constant, k, to the nearest whole number, if a current of 25
amperes falls to 2.5 amperes in 0.01 seconds is ______.
46. Kyle determined the exact solution to the exponential equation 52−𝑥 = 2. He wrote the
answer as the quotient of 2 logarithms in the form
log 𝑀
log 5
. The value of M is ____.
47. A student invests $100 @ 8% per year compounded semi-annually. The amount of
money that the student will have at the end each year is increased from the amount at the
end of the previous year by a factor of:
A.
B.
C.
D.
1.04
1.08
1.0816
1.16
48. George invested $2500in an account which pays compound interest of 8.1% per annum
compounded quarterly. The number of quarters it will take George’s investment to at
least double in value is ________.
49. The tripling period, to the nearest tenth of an hour, of a bacteria culture which grows
from 500 cells to 64000 cells in 50 hours is _____.
50. Radioactive decay material decays to 40% of its original mass in 5 years. The half-life of
the radioactive material, to the nearest hundredth of a year, is ____.
Answer Key:
1) A
2) C
3) A
4) B
5) 5.0
6) 0.36
7) 1.2
8) C
9) D
10) D
11) B
12) C
13) A
14) A
15) C
16) B
17) A
18) 3124
19) 2
20) B
21) A
22) D
23) A
24) A
25) C
26) B
27) D
28) 3.0
29) 729
30) D
31) 7.0
32) 313
33) 4.29
34) B
35) B
36) A
37) D
38) C
39) B
40) A
41) 27
42) A
43) C
44) 0.14
45) 230
46) 12.5
47) C
48) 35
49) 11.3
50) 3.78