1. No category

Exam Questions Sample for Written Test #3
Your exam will have 3 questions picked from the following 6 categories. Each
question might have multiple parts from different categories. The exam questions will be similar in nature but will vary in terms of functions and numbers.
In the exam, the computations will not require a calculator. So you should
attempt to solve these problems without a calculator. Here is a sample of
questions from each category.
1. Graphing
(1.1) Graph the following exponential function. Label all x-intercepts,
y-intercepts, and asymptotes if applicable.
y = −2ex−5 + 3
10
5
-20
-15
-10
-5
0
5
-5
-10
Answer:
1
10
15
20
x-5
y=-2e
4
.
+3
3
y-intercept:
Horizontal Asymptote:
y=3
-5
y=-2e +3
2
1
-1
0
1
2
3
4
5
.
x-intercept:
x= 5 + ln(3/2)
6
7
8
-1
(1.2) Graph the following exponential function. Label all x-intercepts,
y-intercepts, and asymptotes if applicable.
x
1
y =1+3
2
10
5
-20
-15
-10
-5
0
-5
-10
Answer:
2
5
10
15
20
5
4
.
x
y=1+3 (1/2)
y-intercept:
y=4
3
2
1
Horizontal Asymptote:
y=1
-2
-1
0
1
2
3
4
5
6
7
(1.3) Graph the following logarithmic function. Label all x-intercepts,
y-intercepts, and asymptotes if applicable.
y = 1 − ln(x − 4)
10
5
-20
-15
-10
-5
0
-5
-10
Answer:
3
5
10
15
20
y=-ln(x-4)+1
10
5
x-intercept:
x= 4+e
-5
0
5
.
10
15
20
25
30
-5
Vertical Asymptote:
x=4
-10
(1.4) Graph the following logarithmic function. Label all x-intercepts,
y-intercepts, and asymptotes if applicable.
y = log2 (−x + 4) − 3
10
5
-20
-15
-10
-5
0
-5
-10
Answer:
4
5
10
15
20
y=log (-x+4)-3
2
5
Vertical Asymptote:
x=4
x-intercept:
x=-4
-25
-20
-15
-10
-5
. .
0
5
10
y-intercept:
y=-1
-5
-10
-15
(1.5) (HW6 # 3) Determine the correct exponential function of the
form f (x) = bx whose graph is given below.
2.4
2
1.6
1.2
0.8
( 2,
.
0.4
-0.8
-0.4
0
0.4
0.8
1.2
-0.4
Answer: f (x) =
3 x
7
5
1.6
2
-)
9
49
2.4
2.8
3.2
3.6
2. Solving Exponential Equations
(2.1) (HW6 #9) Solve the exponential equation using the method
of ”relating the bases” by first rewriting the equation in the form
bu = bv .
16
1
√
= x
3
x
8
2
(Answer: − 32 )
(2.2) (HW6 #17) Solve the exponential equation using the method
of ”relating the bases” by first rewriting the equation in the form
eu = ev .
2
ex = (e−x ) · e30
(Answer: 5, -6)
(2.3) (HW7 #5) Solve the exponential equation.
153x−1 = 16x−3
(Answer:
−3ln16 + ln15
)
3ln15 − ln16
(2.4) (HW7 #8) Solve the exponential equation.
6(ex−1 )2 · e3x−5 = 204
(Answer:
ln34 + 7
)
5
6
3. Solving Logarithmic Equations
(3.1) (HW6 #58) Solve the logarithmic equation.
log3 (2x + 1) = log3 15
(Answer: x = 7)
(3.2) (HW 6 #59) Solve the logarithmic equation.
ln 4 + ln x = ln 8 + ln(3x − 2)
(Answer: x = 54 )
(3.3) (HW 7 #10) Solve the logarithmic equation.
2 ln x − ln(2x − 9) = ln 7x − ln(x − 2)
(Answer: x = 9, 7)
(3.4) (HW 7 #14) Solve the logarithmic equation.
ln 8 + ln(x2 +
7x
)=0
8
(Answer: x = −1, 81 )
(3.5) (HW 7 #15) Solve the logarithmic equation.
log2 (x − 4) + log2 (x + 5) = 1 + log2 (x)
(Answer: x = 5)
7
4. Word Problems
Note: The answers here were computed with a calculator. In the written exam, we will accept answers non simplified, for example in terms
of ln.
(4.1) (HW 7 #18) What is the interest rate necessary for a investment to triple after 12 years of continuous compound interest? (Round
your answer in percent to two decimal places.)
(Answer: 9.16%)
(4.2) (HW 7 #20) During an experiment, it was found that the
number of bacteria in a culture grew at a rate proportional to its size.
At 9:00 AM there were 3,000 bacteria present in the culture. By noon,
there were 7,200 bacteria. How many bacteria will there be at 6:00
PM?
(Answer: 41472)
(4.3) (HW 7 #21) A radioactive isotope leaked into a small stream.
Eighty-six days after the leak, 20% of the original amount of the substance remained. Determine the half-life of this isotope (round to the
nearest number of days).
(Answer: 37 days)
(4.4) (HW 7 #22) A radioactive isotope is a byproduct of a certain
nuclear reactor. A leak releases a large amount of this isotope into the
environment. Fortunately, the isotope has a short half-life of 8 days.
What is the percentage of the isotope left after 24 days?
(Answer: 12.5%)
(4.5) (HW 7 #25) Police arrive at a murder scene at midnight and
immediately record the temperature of the body at 89◦ F. At 1:00 AM,
the temperature of the body was 84◦ F. If the surrounding temperature
was a constant 70◦ F and the body was 98◦ F at the time of death, when
did the victim die?
(Answer: 10:44 PM)
8
5. Domain of Logarithmic Functions
(5.1) (HW6 #41) Find the domain of the logarithmic function.
f (x) = log8 (3x + 2)
(Answer: (− 23 , ∞))
(5.2) (HW 6 #42) Find the domain of the logarithmic function.
f (x) = log 1 (x − 3)
3
(Answer: (3, ∞))
(5.3) (HW 6 #43) Find the domain of the logarithmic function.
f (x) = ln(5x − 32) + 32
(Answer: ( 32
, ∞))
5
(5.4) (HW 6 #45) Find the domain of the logarithmic function.
f (x) = log(x2 + 4x − 12)
(Answer: (−∞, −6) ∪ (2, ∞, ))
(5.5) (HW 6 #46) Find the domain of the logarithmic function.
f (x) = ln(−x2 + 4x − 3)
(Answer: (1, 3))
9
6. Properties of Logarithms and Exponentials
(6.1) (HWK 6, #24) Convert to a logarithmic equation.
3−2 =
(Answer: log3
1
9
1
9
= −2.)
(6.2) (HWK 6, #29) Evaluate the logarithm without the use of a
calculator.
1
log5 √
5
125
(Answer:
−3
.)
5
(6.3) (HWK 6, #32) Use the properties of logarithms to evaluate
the expression without the use of a calculator.
loga 1,
a>1
(Answer: 0.)
(6.4) (HWK 6, #37) Write the logarithmic equation as an exponential equation.
log M = B
(Answer: M = 10B )
(6.5) (HWK 6, #55) Use the properties of logarithms to expand
the logarithmic expression. Wherever possible, evaluate logarithmic
expressions.
√
2x4
log2 p
5
8y 4
1
(Answer: − 10
+ 2 log2 x − 45 log2 y)
10