Download MTH-112 Quiz 12

MTH-112 Quiz 12
Name:
#:
Please write your name in the provided space. Simplify your answers. Show your work.
2012
> 0. Graph the solution set on a
1. Solve
x2
real number line. Write the solution set in interval notation.
-6 -5 -4 -3 -2 -1 0
1
2
3
4
5
3. Use the log function f (x) = log2 (x − 1) to find
the following.
(a) The domain (write in interval notation):
6
(b) The range (write in interval notation):
(c) The equation of the vertical asymptote:
(d) The equation of the horizontal asymptote:
2. Use the exponential function f (x) = −2x+1 to
find the following.
4. Using the definition of logarithms, find the exact value without using a calculator.
(a) The domain (write in interval notation):
(a) log1234 12342
(b) log5 25
(b) The range (write in interval notation):
(c) 4log4 16
(c) The equation of the vertical asymptote:
(d) 3log3 25
(d) The equation of the horizontal asymptote:
(e) log4
1
1
16
MTH-112 Quiz 12 - Solutions
Words in italics are for explanation purposes only (not necessary to write in the tests or
quizzes).
2012
> 0. Graph the solution set on a
1. Solve
x2
real number line. Write the solution set in interval notation.
(a) The domain (write in interval notation):
The domain of any exponential function is
always all real numbers.
Since the numerator is just a number, it does not
have zeros. Find the zeros of the denominator
x(x2 + 1) = 0.
(−∞, ∞)
(b) The range (write in interval notation):
The range of f (x) = 2x is (0, ∞).
x2 = 0
The range of f (x) = −2x is (−∞, 0), because the negative sign flips the graph (and
range) of f (x) = 2x over x−axis.
x=0
Separate the number line into two intervals using
the point x = 0.
-6 -5 -4 -3 -2 -1 0
1
2
3
4
5
The +1 in f (x) = −2x+1 shifts the graph
of f (x) = −2x left one unit. Since the left
(or right) shift does not affect the range, the
range of f (x) = −2x+1 is the same as that
of f (x) = −2x , which is
(−∞, 0)
6
(c) The equation of the vertical asymptote:
Pick two test points, one from each interval, and
check whether the factored form of the polynomial
is a positive or negative value. We are interested
in positive values, because we are looking for ≥ 0.
There are no vertical asymptotes in the
graphs of exponential functions.
None
Test
(d) The equation of the horizontal asymptote:
Pts
2012
x2
>0
−1
2012
=
(−1)2
(+)
= (+)X
(+)
2012
=
(1)2
(+)
= (+)X
(+)
1
?
The equation of the horizontal asymptote of
an exponential function is always y = the
number used when writing the range.
y=0
3. Use the log function f (x) = log2 (x − 1) to find
the following.
(a) The domain (write in interval notation):
Both test points 1 and −1 yield positive values.
×
-6 -5 -4 -3 -2 -1 0
The −1 in f (x) = log2 (x − 1) shifts the
graph (and the domain) of f (x) = log2 x
right one unit. Therefore, the domain is
X
1
2
3
4
5
6
(1, ∞)
(b) The range (write in interval notation):
The interval notation: (−∞, 0) ∪ (0, ∞)
The range of any logarithmic function is always all real numbers.
2. Use the exponential function f (x) = −2x+1 to
find the following.
(−∞, ∞)
1
MTH-112 Quiz 12 - Solutions
(c) The equation of the vertical asymptote:
4. Using the definition of logarithms, find the exact value without using a calculator.
The equation of the vertical asymptote of a
logarithmic function is always x = the number used when writing the domain.
(a) log1234 12342
= 2 · log1234 1234 = 2 · 1 = 2
x=1
(b) log5 25 = 2
(d) The equation of the horizontal asymptote:
(c) 4log4 16 = 16
There are no horizontal asymptotes in the
graphs of logarithmic functions.
(d) 3log3 25 = 25
1
1
= log4 4−2
(e) log4
= log4
2
16
4
= −2
None
2
MTH-112 Quiz 13
Name:
#:
Please write your name in the provided space. Simplify your answers. Show your work.
(
√ )
125 x
1. Using the definition of logarithms, find the ex(e) log5
y4
act value without using a calculator.
(a) 4log2 16
.
( )
(b) log4 163
.
3. Condense the logarithmic expression as much as
possible. Where possible, evaluate logarithmic
expressions without using a calculator.
.
2. Expand the logarithmic expression as much as
possible. Where possible, evaluate logarithmic
expressions without using a calculator.
(a) 9 logb x + 4 logb y
(a) log3 (81x2 )
.
.
(
(b) log4
256
x2
(b) 4 logb 10 − 2 logb 100
)
.
.
(
(c) log4
256x
y2
(c) log 250 + log 4
)
.
.
(d) log5
(
125x2
y4
)
4. Solve: 3x−2 = 27
.
.
1
MTH-112 Quiz 13 - Solutions
Words in italics are for explanation purposes only (not necessary to write in the tests or
quizzes).
1
= log5 125 + log5 x 2 − log5 y 4
1
= 3 + log5 x − 4 log5 y
2
1. Using the definition of logarithms, find the exact value without using a calculator.
(a) 4log2 16 = 44 = 256
(b) log4 163 = 3 · log4 16 = 3 · 2 = 6
3. Condense the logarithmic expression as much as
possible. Where possible, evaluate logarithmic
expressions without using a calculator.
2. Expand the logarithmic expression as much as
possible. Where possible, evaluate logarithmic
expressions without using a calculator.
(a) 9 logb x + 4 logb y
= logb x9 + logb y 4
= logb (x9 y 4 )
(a) log3 (81x2 )
= log3 81 + log3 x2
= 4 + 2 log3 x
256
(b) log4
x2
= log4 256 − log4 x2
= 4 − 2 log4 x
256x
(c) log4
y2
= log4 256 + log4 x − log4 y 2
= 4 + log4 x − 2 log4 y
125x2
(d) log5
y4
= log5 125 + log5 x2 − log5 y 4
= 3 + 2 log5 x − 4 log5 y
√ 125 x
(e) log5
y4
!
1
125x 2
= log5
y4
(b) 4 logb 10 − 2 logb 100
= logb 104 − logb 1002
4 10
= logb
1002
10000
= logb
10000
= logb 1
=0
(c) log 250 + log 4
= log(250 · 4)
= log 1000
=3
4. Solve: 3x−2 = 27
3x−2 = 33
x−2=3
x=5
1
MTH-112 Quiz 14
Name:
#:
Please write your name in the provided space. Simplify your answers. Show your work.
(c) log2 (x − 2) + log2 (x + 1) = 2
Solve the following equations. Write the exact answers using natural logarithms if necessary. Then use
a calculator to obtain decimal approximations (round
to three decimal places).
1. 4x+3 = 128
.
.
2. 10x = 2.22
(d) log3 (x + 24) − log3 (x − 2) = 3
.
3. 6ex = 120.5
.
.
4. Solve the following equations. Write the exact
answers. Then use a calculator to obtain decimal approximations (round to three decimal
places).
5. A = 739.4e0.01t represents the population, A, of
a country in millions, t years after 2003. When
will be the population of the country 959 millions?
(a) ln x = 7
.
(b) log3 (x − 5) = 2
.
.
1
MTH-112 Quiz 14 - Solutions
Words in italics are for explanation purposes only (not necessary to write in the tests or
quizzes).
(c) log2 (x − 2) + log2 (x + 1) = 2
Solve the following equations. Write the exact answers using natural logarithms if necessary. Then use
a calculator to obtain decimal approximations (round
to three decimal places).
log2 (x − 2)(x + 1) = 2
(x − 2)(x + 1) = 22
x2 − x − 2 = 4
1. 4x+3 = 128
x+3
= 27
22
x2 − x − 6 = 0
(x − 3)(x + 2) = 0
22(x+3) = 27
x = 3, −2
2(x + 3) = 7
Since logarithm of a negative number is undefined, x 6= −2. (−2 makes log2 (x + 1)
undefined.)
2x + 6 = 7
2x = 1
1
x=
2
(d) log3 (x + 24) − log3 (x − 2) = 3
2. 10x = 2.22
ln(10x ) = ln 2.22
log3
x ln 10 = ln 2.22
ln 2.22
x=
≈ 0.346
ln 10
x + 24
=3
x−2
x + 24
= 33
x−2
x + 24 = 27(x − 2)
x + 24 = 27x − 54
3. 6ex = 120.5
120.5
ex =
6
120.5
x
ln(e ) = ln
6
120.5
≈3
x = ln
6
78 = 26x
3=x
6. A = 739.4e0.01t represents the population, A, of
a country in millions, t years after 2003. When
will be the population of the country 959 millions?
A = 739.4e0.01t
5. Solve the following equations. Write the exact
answers. Then use a calculator to obtain decimal approximations (round to three decimal
places).
959 = 739.4e0.01t
959
= e0.01t
739.4
959
ln
= ln e0.01t
739.4
959
ln
= 0.01t
739.4
959
ln 739.4
=t
0.01
t ≈ 26
(a) ln x = 7
x = e7 ≈ 1096.633
(b) log3 (x − 5) = 2
x − 5 = 32
x−5=9
x = 14
In 2029.
1
MTH-112 Quiz 15
Name:
#:
Please write your name in the provided space. Simplify your answers. Show your work.
1
(e) (Multiply
) the second row by − 5
1
− 5 R2 . This will give you 1 as the second entry of the second row.
1. Solve the system using matrices.
4x − y + z = 0
x + y − z = −5
−x + 3y − 2z = 1
(a) Write the augmented matrix.
.
(f) Multiply the second row by −4 and add it
to the third row (−4R2 + R3 ). This will
give you 0 as the second entry of the third
row.
.
(b) Interchange the first and second rows
(R1 ←→ R2 ). This will give you 1 as the
first entry of the first row.
.
(g) Write the resulting system of linear equations.
.
(c) Multiply the first row by −4 and add it
to the second row (−4R1 + R2 ). This will
give you 0 as the first entry of the second
row.
.
(h) Solve the system by back substituting.
.
(d) Multiply the first row by 1 and add it to
the third row (1R1 + R3 ). This will give
you 0 as the first entry of the third row.
.
.
1