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Sample Test Questions
(Barnett, Ziegler, and Byleen)
Chapter 5
Exponential and Logarithmic Functions
5-1
exponential functions
1.
How does the graph of f(x)
2.
What is the value of all exponential functions at x
3.
The population of Texas grows at the rate of 3% increase each year. If the population were
16,400,000 in 1985, then what would be the population in 2012?
(5)
4.
An investment grows exponentially in such a way that it doubles in 10 years. If $1000 were
invested initially, then what is the amount after 7 years?
(5)
5-2
the exponential function with base e
5.
Simplify
6.
Sketch the graph for:
x
x
= 2 relate to the graph of g(x) = (1/2) ?
= 0?
e 4 x e"3x
e 2x
(3)
(5)
"x / 2
! y = 5e , x # 0
7.
(5)
(5)
The spread of a certain epidemic is governed by:
!
N(t) =
100000
1+ 999e"0.5t
where N(t) is the number that have been infected in the population after t days.
what is the initial number infected; that is, N(t=0)?
! a.
b.
what value does
asymptote?
N approach as t gets large; that is, what is the horizontal
(5)
sketch the graph of this function for 0
c.
≤ t ≤ 20 days
5-3
logarithmic functions
8.
How does the graph of f(x)
9.
What is the value of all logarithmic functions at x
10.
If e
x
(5)
x
= 2 relate to the graph of g(x) = log2 x?
= 3, then what is the value of x?
= 1?
(5)
(5)
(5)
(5)
2
11.
If ln
12.
Simplify; that is, write as a single log function or simpler:
13.
x = −1, then what is the value of x?
1
z
a.
2log x + log y " log
b.
ln(x 2 " 4) " ln(x " 2) + ln(x + 2)"1
(5)
(5)
(5)
! Compute the values of the following. Hint: you do not need a calculator.
ln e5
(3)
b.
log 1000
(3)
c.
log3
d.
9log3 5
(3)
e.
log3 94
(3)
! a.
34
5-4
common and natural logarithms
14.
Compute the following values to 4 digits to the right of the decimal place.
(3)
a.
ln 0.5
(3)
b.
5.343.1415
(3)
c.
log2 25.5
(3)
d.
log e4
(3)
e.
e−1.3
(3)
f.
ln −0.5
(3)
g.
log3 52
(3)
h.
3.75−2.1
(3)
I
log7 25.5
(3)
j.
ln 102
(3)
k
log 0.02
(3)
l.
log e3
(3)
3
15.
16.
Tritium, a radioactive isotope of hydrogen, has a half-life of 12.33 years. It is used, when
mixed with a phosphorescence substance, to illuminate watch dials, rifle sights, etc. for night
use. If 0.10 milligrams is used, how much remains after
a.
5 years
(5)
b.
20 years
(5)
It is determined that a wooden beam taken from the ruins of a house has 30.7% of the
14
14
amount of C as a freshly cut log. The amount of C is in equilibrium when the entity is
14
living. When it dies, the amount of C decays with a half-life of 5730 years. That is:
A(t) = A0e−ln2 t/H
Where the amount at time t, A(t), is calculated from the initial amount A0, and the half-life, H.
How long has it been since the wooden beam was harvested from its tree and incorporated
into the structure of the house?
(10)
17.
The atmospheric pressure P, in pounds per square inch, decreases exponentially with
altitude h, in miles (5280 ft/mi) above sea level, as given by
P(h) = 14.7 e−0.21h
18.
a.
What is the air pressure, in pounds per square inch, at the peak of Mt. Everest which
is 29,032 feet?
(5)
b.
What is the air pressure, in pounds per square inch, at Keene, TX with an elevation
of 960 feet?
(5)
c.
What is the air pressure, in pounds per square inch, at the Dead Sea which is 1312
feet below sea level?
(5)
The sound level of a sound, D, is measured in decibels (db) what are defined according to:
"I%
D(I) = 10log$ '
# I0 &
where I is the sound intensity and I0
19.
= 10−16 watts per square centimeter.
14
! a.
If a whisper has a sound intensity of 5.2x10− watts per square centimeters, then
what is the sound level of a whisper in decibels?
(5)
b.
If the noise of rustling leaves has a sound level of 10 db, then what is the sound
intensity in watts per square centimeter.
(5)
The specification for an audio amplifier states that the signal-to-noise ratio is −50 db. This
means the sound level of the noise (hiss and static) in the system is 50 db less than the
sound level of the normal music. This difference in sound level corresponds to what ratio in
sound intensities of the noise to the music?
(5)