Review Problems Quarterly #2 Exam
Pre-Calculus Chapter 3 and 4.1 – 4.3
Name: ____________________
Chapter 3: Exponential and Logarithmic Functions
Properties of Logarithms
a = bx
is the same as
log b a = x
log a 1 = 0 log a a = 1 log a a x = x
If log a x = log a y, then x = y
1.
Write the exponential equation in logarithmic form.
1
64 2 / 3 = 16
5−1 =
a)
b)
5
2.
Evaluate without the use of a calculator.
a)
log 3 27
Change of Base Formula
log b x log x ln x
log a x =
=
=
log b a log a ln a
3.
b)
log10 0.001
c)
log 5
1
25
where a ≠ 1, b ≠ 1, and a, b, x > 0
Evaluate with your calculator. Remember to use the change of base formula.
a)
log 5 2
b)
log 3 25
c)
ln17
Graph each equation.
4.
y = 2x
y = log 2 x
5.
Properties of Logarithms (Same properties apply for ln)
For all, b ≠ 1 and b is positive, p ∈ Real Numbers and m and n are > 0
Product Property:
logb mn = logb m + logb n
m
Quotient Property: logb = logb m − logb n
n
Power Property:
log b m p = p ⋅ log b m
6.
Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple
of logarithms.
a)
7.
log 7 4 x 3 y
b)
⎛ x − 2⎞
ln⎜ 2 ⎟
⎝ 5x ⎠
c)
⎛ 3y x ⎞
log 2 ⎜
⎟
⎝ 4 ⎠
Use the properties of logarithms to condense the expression to the logarithm of a single quantity.
a)
5[ln(3x −1) + ln2 + 2ln(x −1)]
b)
1
3log x + log(x + 7) − log(3x − 7)
3
8.
Solve for x.
2 x = 256
b)
3ln x = 7
c)
3x =
d)
e x = 14
e)
3e1.2 x + 5 = 17
f)
−6(2 x ) = −15
h)
ln x − ln 7 = 1
i)
log 3 x + log 3 (x −1) = log 3 (x + 1)
g)
9.
1
81
a)
5 i 56 = 52 x
⎛ 3 ⎞t
After t years, the value of a car that costs $26,000 is modeled by V (t) = 26,000⎜ ⎟ .
⎝ 4⎠
a)
Use a graphing utility to graph the function.
b)
Find the value of the car 2 years after it was purchased.
c)
According to the model, when does the car depreciate most rapidly? Is this realistic? Explain.
10.
You deposit $7550 into an account that pays 7.25% interest, compounded continuously. How long will
nt
r⎞
⎛
it take for the money to triple? (Choose the correct formula: A = P ⎜ 1+ ⎟ or A = Pert )
⎝ n⎠
11.
Determine the nearest positive and negative coterminal angles. When the angle is given in degrees, give
your answer in degrees. When the angle is given in radians, give your answer in radians.
a)
-85º
b)
5π
4
c)
−
π
4
12.
A record rotates at a rate of 45 revolutions per minute. If the radius of the record is 6 in, what is its
linear velocity (in inches/min)?
13.
The moon rotates once around the earth every 27 days. What is its angular velocity? If its orbital radius
is 385000km, what is its linear velocity in km/hr?
14.
Determine the linear velocity of a person riding the Gravitron in Wildwood. The radius of the Gravitron
is 20 ft and it rotates at a rate of 24 rpm.
15.
Convert to radians.
a)
16.
135º
b)
-330º
c)
5º
b)
π
5
c)
4 radians
Convert to degrees.
a)
1.65
17. Convert each to decimal degrees.
a.) 250 15’ 30”
b.) – 1240 59”
18. Convert to DMS.
a.) 58.47630
b.) – 324.76380
19.
A circle has radius of 5 cm. Find the length of the arc subtended by a central angle of 50º.
20.
Find the length of the arc of a circle subtended by a central angle of
is 3.9 cm.
5π
radians if the radius of the circle
6
21.
What angle is subtended by an arc length of 12 inches in a circle with radius 2.7 inches? Give your
answer in radians and degrees.
22.
If the area of a circular sector is 100 square feet, determine the length of the arc that is subtended if the
circle has a radius of 10 ft.
23.
Determine the distance between two cities that lie on the same longitudinal line on the same side of the
equator. The first city has a latitude of 240 28’30” and the second city has a latitude of 300 24’28”
(Use r = 3964 miles).
24.
Name the reference angle for each angle given below.
a)
125º
b)
-70º
25.
If θ = 27!15'28" , then tan θ = ?
26.
If sin θ = 2/3 and terminates in Quadrant I, then cos θ = ?
27.
Determine which quadrant each angle terminates in.
a)
-188º
b)
272º
c)
5π
6
c)
7π
6
28.
If tan θ > 0 and sin θ < 0, then θ terminates in quadrant _________.
29.
The point (-3,7) is on the terminal side of angle θ . Find the exact value of all six trig functions of
angle θ .
30.
Given sec θ = -2 and sin θ > 0, determine the exact value of the remaining trig functions.
31.
Given sin θ = −
32.
Find the EXACT value of each of the following. (NO CALCULATOR!! ~ Draw a ref. triangle)
3
and tan θ > 0, determine the exact value of the remaining trig functions.
2
a)
cos 0º
b)
sin 270º
c)
tan 45º
d)
sin (-30º)
e)
tan (-60º)
f)
cot
π
3
5π
2
g)
sec 150º
h)
csc 225º
i)
cos
j)
⎛ 15π ⎞
sin ⎜ −
⎝ 4 ⎟⎠
k)
⎛ 5π ⎞
cos ⎜ − ⎟
⎝ 6 ⎠
l)
sec 390º
33.
A ladder 3.0 m long is placed against a building at an angle of 60º with the ground. How high up the
wall will the ladder reach?
34.
To find the height of a pole, a surveyor moves 100 feet from the base of the pole and then, from an eyelevel height of 6.5 feet, measures the angle of elevation to the top of the pole to be 40º. Find the height
of the pole to the nearest foot.
35.
The cable supporting a ski lift rises 3 feet for every 8 feet of horizontal length. The top of the cable is
fastened 675 feet above the cable’s lowest point. Find the lengths b and c, and find the measure of ∠ A .
c
675
feet
A
b