Alpha Logs and Exponents
MAΘ National Convention 2012
For this test, the acronym NOTA stands for “None of These Answers.” Unless otherwise stated, assume
that i 
1 . f
1
x  denotes the inverse function of f x .
1. Which of the following is equivalent to e 5 ln x ( x  0 )?
B. x  5
A. 5
2.



log 4 log 3 log 2 ln e 8
A. -1
C. 5 x
D. x 5
E. NOTA
C. 1
D. 2
E. NOTA
  ?
B. 0
3. Solve for x: 7 5 x  4  6 3 x 5
A.
4 ln 7  5 ln 6
3 ln 6  5 ln 7
4. Given that f x   n
A.
ln x
ln n
x
B.
ln 6
ln 7
C.
ln 7
ln 6
4 ln 7  5 ln 6
5 ln 7  3 ln 6
D.
n  1, which of the following is
B. log nx 
C. n log x
f
1
E. NOTA
x  ?
D. ln nln x 
E. NOTA
5. The amount of points scored by a football team is directly proportional to the logarithm of the
quarterback’s passer rating and inversely proportional to the logarithm of the team’s rushing
average. Given that a team scores 21 points with a passer rating of 64 and a rushing average of
4, how many points will they score with a passer rating of 128 and a rushing average of 2?
63
A. 2
147
B. 4
C. 42
D. 49
E. NOTA
6. Given that a log 6 3  b log 6 2  c log 6 5  5 , for the rational numbers a, b and c which of the
following must be true?
A. a  b
B. a  b
7.
C. b  a
D. c  a  b
E. NOTA
56  56  56    ?
A. -8
B. 7
C.
15
2
Page 1 of 6
D. 8
E. NOTA
Alpha Logs and Exponents
MAΘ National Convention 2012
8. As a rocket lifts off into space, its total mass (in kg) at time t seconds after liftoff can be
represented by mt   mo e

a
t
vo
where m o is the initial mass of the rocket, v o is the rocket’s
initial velocity and a is the acceleration of the rocket (which is assumed to be constant for this
problem). Given that the rocket’s initial velocity is ten times its acceleration, find the time in
seconds when the rocket has a mass of 500 kg.
A. 10ln 500  ln mo  B.
1
ln 500  ln mo  C. 1 ln mo  ln 500 D. 10ln mo  ln 500
10
10
E. NOTA
9. Allow f  x  

 cos
n 0
A.
1
4
2n
 
x . Find f   .
3
3
B.
4
C.
4
3
D. 4
E. NOTA
10. Find the area of a triangle with two sides of length log 49 512 and log 16 343 with the angle
between the two sides being
A.
27
32

.
3
B.
27 3
32
C.
27
16
D.

11. Find the sum of the solutions to log 3 x  5  log 9 x 2  3x  10
A. -5
B. 0
27 3
16
E. NOTA

C. 2
D. 5
E. NOTA
12. Which of the following gives a correct value of ln  81?
A.  4 ln 3
B.  4i
C. 4 ln 3  i
Page 2 of 6
D. 4 ln 3 
3
i
2
E. NOTA
Alpha Logs and Exponents
MAΘ National Convention 2012
13. Estimate the area under the graph of f x   log 2 x and above the x-axis by dividing the interval
between x  1 and x  16 equally into the widths of five rectangles with the upper right vertex
of each rectangle on the graph of f x  so that the value of the height of each rectangle is the
value of the function at the x coordinate of the right endpoint of each width.
A. 7  log 2 455 B. 14  log
2
455 C. 21  log 2 455 D. 21  log 3 2 455
E. NOTA
 x 2  3x  2 
 x  0?
 x 2  5x  6 


B. log 4 x  1  log 4 x  2  log 4 x  3
14. Which of the following is an expansion of log 4 
A. log 2 x  1  log 2 x  2  log 4 x  3
C. log 16 x  1  log 4 x  2  log 4 x  3 D. log 16 x  1  log 16 x  2  log 4 x  3
E. NOTA
15. Find the sum of the real roots of y  x  73 x 2  73 x  15.
A. 7
B. 15
C. 151
D. 153
E. NOTA
16. Find the function that results from shifting f x   e 2 x 2 to the right by
3
and then reflecting it
2
over the line y  x .
A. y  e 2 x 1
B. y  e12 x
C. y 
ln x  5
ln x  1
D. y 
2
2
17. Given that a  log 81 3 and b  log 7 9 , which of the following equals log 9
A. a  b
B. a  b
C.
ab  1
b
Page 3 of 6
D.
aba  b 
ab
E. NOTA
3
?
7
E. NOTA
Alpha Logs and Exponents
MAΘ National Convention 2012
18. Given that f x  is an exponential function, how many of the following must be true?
I.
II.
f x   0
f x  is always increasing on its domain.
III. log  f x  is a linear function.
IV. If the domain of f x  is  , 0 , then lim f  x   0.
x  
V. The range of  f  x 
sin x
 1 f  x 
,
 for the domain given by  1000  x  1000
 f x  1 
is 
A. 0
B. 1
C. 2
D. 3
E. NOTA
19. The parametric equations x  7 t and y  7  t are part of which of the following graphs?
A. ellipse
B. hyperbola
C. line
D. parabola
E. NOTA
20. For the following four statements, the value in parentheses is positive if the statement is true
and negative if it is false. Find the sum of the values.
log x  ln x for all x that lie in both domains. (3)
I.
x
II.
1

lim 1    e (5)
x 
x

III.
y
IV.
 xm
ln  n
y
x log x
3  ln x
is continuous over the real numbers excluding zero. (7)

  m ln x  n ln y (given that x and y are positive) (4)

A. -1
B. 3
C. 5
21. Which of the following is a fifth root of
A. 
22. e
2 3
i
arc csc

 3 
3 2 3 2i

2
2
B. 
D. 19
E. NOTA
243 2 243 2

i?
2
2
3 3 3i

2
2
C.
3 3 3i

2
2
D.
3 2 3 2i

2
2
E. NOTA
?
A. 
1
3

i
2
2
B.
1
3

i
2
2
C.
Page 4 of 6
1
3

i
2
2
D.
3 i

2
2
E. NOTA
Alpha Logs and Exponents
23. Given that log x 
MAΘ National Convention 2012
1
2
and log y  , find the value
3
5

 






of log sin arccos 1  x 2  log  cos arctan
1 x2 y2
xy

  . (Assume that the arccosine and


arctangent functions have their traditional ranges)
2
5
A. 
1
3
B. 
C.
1
3

D.
2
5
E. NOTA

24. For this problem, consider the set 2,2 2 ,2 3 ,,2 24 ,2 25 . Two numbers, a and b, are chosen
randomly from this set with a being replaced in the set before b is chosen. What is the
probability that log a b is an integer?
A.
11
150
B.
87
625
C.
88
625
D.
29
100

25. Find the sum of all the distinct values of x such that x 2  11x  27
A. -18
B. -29
C. -31
E. NOTA

x 2  7 x 10
D. -33
 1.
E. NOTA
26. Solve for x: log 5 27  ln 5 x  ln 81
A. log 3
3
4
B.
C. ln 3
D.
4
3
E. NOTA
7
 log1 8n x
27. Which of the following equals x n 1
A.
1
8!
x
B.
x
C. 7!
1
7!
?
x
x
D. 8!
x
E. NOTA
28. Find the half-life of a substance that decays exponentially in respect to the
function f t   Ce  kt . (Leave your answer in terms of the constants C and k )
A. ln 2
 1 
k 
2 
B. ln 
C. ln 2 k
Page 5 of 6
D. ln k
E. NOTA
Alpha Logs and Exponents
29. Consider the function f  x  
MAΘ National Convention 2012
5  2e x
which has a horizontal asymptote at y  a and a
e x  ln  x  e 
rational vertical asymptote at x  b . Find a  b .
A. -2
B. 1
C. 2
D. 3
E. NOTA
63
30.
 log n  1 (hint- the symbol denotes the product of the values, not the sum)
n
n4
A. log 63 64
B. log 4 63
C. 3
Page 6 of 6
D. 16
E. NOTA