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Math Problems
Algebra 2
Product property of logarithms
14
14
14
\sqrt{
5
5
5
}+
3
3
3
\sqrt{
15
15
15
}+
3
3
3
\sqrt{
5
5
5
}+
12
12
12
\sqrt{
15
15
15
}
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Simplify each logarithm
\newline
log
3
7
+
log
3
8
\log_{3}7+\log_{3}8
lo
g
3
7
+
lo
g
3
8
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Simplify the expression:
8
+
(
5
3
)
+
x
+
3
=
□
8+(\frac{5}{3})+x+3 = \square
8
+
(
3
5
)
+
x
+
3
=
□
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Make the equation true.
\newline
□
×
□
+
(
□
÷
2
)
=
15
\square \times \square +( \square \div 2)=15
□
×
□
+
(
□
÷
2
)
=
15
\newline
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Which two expressions are equivalent to each other?
\newline
Multi-select Choices:
\newline
(A)
7
6
×
7
2
7^6 \times 7^2
7
6
×
7
2
\newline
(B)
1
7
8
\frac{1}{7^8}
7
8
1
\newline
(C)
7
11
7^{11}
7
11
\newline
(D)
7
12
7
\frac{7^{12}}{7}
7
7
12
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Select all the expressions that are equivalent to
6
2
2
2
\frac{6^2}{2^2}
2
2
6
2
.
\newline
Multi-select Choices:
\newline
(A)
3
0
3^0
3
0
\newline
(B)
1
3
−
2
\frac{1}{3^{-2}}
3
−
2
1
\newline
(C)
3
2
3^2
3
2
\newline
(D)
1
3
2
\frac{1}{3^2}
3
2
1
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Evaluate the expression
5
⋅
x
3
x
2
\frac{5 \cdot x^{3}}{x^{2}}
x
2
5
⋅
x
3
for
x
=
2
x=2
x
=
2
\newline
□
\square
□
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Simplify to a single trig function with no denominator.
\newline
csc
2
θ
⋅
cos
2
θ
\csc ^{2} \theta \cdot \cos ^{2} \theta
csc
2
θ
⋅
cos
2
θ
\newline
Answer:
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Condense the logarithm
\newline
x
log
b
+
7
log
k
x \log b+7 \log k
x
lo
g
b
+
7
lo
g
k
\newline
Answer:
log
(
□
)
\log (\square)
lo
g
(
□
)
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Simplify
3
y
4
⋅
2
y
⋅
y
4
3 y^{4} \cdot 2 y \cdot y^{4}
3
y
4
⋅
2
y
⋅
y
4
=
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Problem
\newline
What is the product of all solutions to the equation
\newline
log
7
x
2023
⋅
log
289
x
2023
=
log
2023
x
2023
\log _{7 x} 2023 \cdot \log _{289 x} 2023=\log _{2023 x} 2023
lo
g
7
x
2023
⋅
lo
g
289
x
2023
=
lo
g
2023
x
2023
\newline
(A)
(
log
2023
7
⋅
log
2023
289
)
2
\left(\log _{2023} 7 \cdot \log _{2023} 289\right)^{2}
(
lo
g
2023
7
⋅
lo
g
2023
289
)
2
\newline
(B)
log
2023
7
⋅
log
2023
289
\log _{2023} 7 \cdot \log _{2023} 289
lo
g
2023
7
⋅
lo
g
2023
289
\newline
(C)
1
1
1
\newline
(D)
log
7
2023
⋅
log
289
2023
\log _{7} 2023 \cdot \log _{289} 2023
lo
g
7
2023
⋅
lo
g
289
2023
\newline
(E)
(
log
7
2023
⋅
log
289
2023
)
2
\left(\log _{7} 2023 \cdot \log _{289} 2023\right)^{2}
(
lo
g
7
2023
⋅
lo
g
289
2023
)
2
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Factorise.
x
2
−
2
x
+
1
=
0
x^2 - 2x + 1 = 0
x
2
−
2
x
+
1
=
0
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Express as a single logarithm.
\newline
3
log
a
3
+
7
log
a
5
3\log_{a}3+7\log_{a}5
3
lo
g
a
3
+
7
lo
g
a
5
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(
log
(
x
−
2
)
3
y
)
−
64
−
x
2
−
y
2
\left(\frac{\log(\sqrt{x}-2)}{3y}\right)-\sqrt{64-x^{2}-y^{2}}
(
3
y
lo
g
(
x
−
2
)
)
−
64
−
x
2
−
y
2
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Express the given expression without logs, in simplest form. Assume all variables represent positive values.
\newline
(
1
1
log
11
(
7
w
)
−
log
11
(
9
y
2
)
)
\left(11^{\log _{11}(7 \sqrt{w})-\log _{11}\left(9 y^{2}\right)}\right)
(
1
1
l
o
g
11
(
7
w
)
−
l
o
g
11
(
9
y
2
)
)
\newline
Answer:
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Find the volume of a sphere with a radius of
4
4
4
feet.
\newline
Leave
π
\pi
π
\newline
(A)
341.33
π
f
t
3
341.33 \pi \mathrm{ft}^{3}
341.33
π
ft
3
\newline
(B)
16
π
f
t
3
16 \pi \mathrm{ft}^{3}
16
π
ft
3
\newline
(C)
21.33
π
f
t
3
21.33 \pi \mathrm{ft}^{3}
21.33
π
ft
3
\newline
(D)
85.33
π
f
t
3
85.33 \pi \mathrm{ft}^{3}
85.33
π
ft
3
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Simplify. Your answer should be in proper scientific notation:
\newline
(
2.5
×
1
0
4
)
(
4
×
1
0
3
)
\left(2.5 \times 10^{4}\right)\left(4 \times 10^{3}\right)
(
2.5
×
1
0
4
)
(
4
×
1
0
3
)
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Click and drag like terms onto each other to simplify fully.
\newline
5
x
+
4
x
2
−
6
+
6
+
2
y
+
x
+
4
5 x+4 x^{2}-6+6+2 y+x+4
5
x
+
4
x
2
−
6
+
6
+
2
y
+
x
+
4
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Click and drag like terms onto each other to simplify fully.
\newline
−
5
y
2
+
1
+
3
y
2
−
5
x
+
3
y
2
−
5
y
3
+
7
y
3
-5 y^{2}+1+3 y^{2}-5 x+3 y^{2}-5 y^{3}+7 y^{3}
−
5
y
2
+
1
+
3
y
2
−
5
x
+
3
y
2
−
5
y
3
+
7
y
3
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Condense each expression to a single logarithm.
\newline
2
log
6
u
−
8
log
6
v
2 \log _{6} u-8 \log _{6} v
2
lo
g
6
u
−
8
lo
g
6
v
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Find all values of
x
x
x
.
\newline
log
2
(
2
x
2
+
2
)
−
log
2
(
3
x
+
1
)
=
0
\log_{2}(2x^{2}+2)-\log_{2}(3x+1)=0
lo
g
2
(
2
x
2
+
2
)
−
lo
g
2
(
3
x
+
1
)
=
0
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Evaluate the expression
\newline
(
5
x
3
)
/
(
x
2
)
(5x^{3})/(x^{2})
(
5
x
3
)
/
(
x
2
)
for
x
=
2
x=2
x
=
2
\newline
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Which of the following functions are continuous at
x
=
2
x=2
x
=
2
?
\newline
f
(
x
)
=
x
−
4
4
h
(
x
)
=
log
(
x
−
4
)
\begin{array}{l} f(x)=\sqrt[4]{x-4} \\ h(x)=\log (x-4) \end{array}
f
(
x
)
=
4
x
−
4
h
(
x
)
=
lo
g
(
x
−
4
)
\newline
Choose
1
1
1
answer:
\newline
(A)
f
f
f
only
\newline
(B)
h
h
h
only
\newline
(C) Both
f
f
f
and
h
h
h
\newline
(D) Neither
f
f
f
nor
h
h
h
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5
⋅
7
2
y
=
175
5 \cdot 7^{2 y}=175
5
⋅
7
2
y
=
175
\newline
Which of the following is the solution of the equation?
\newline
Choose
1
1
1
answer:
\newline
(A)
y
=
log
7
(
35
)
y=\log _{7}(35)
y
=
lo
g
7
(
35
)
\newline
(B)
y
=
log
35
(
7
)
2
y=\frac{\log _{35}(7)}{2}
y
=
2
l
o
g
35
(
7
)
\newline
(C)
y
=
log
7
(
35
)
2
y=\frac{\log _{7}(35)}{2}
y
=
2
l
o
g
7
(
35
)
\newline
(D)
y
=
log
35
(
7
)
y=\log _{35}(7)
y
=
lo
g
35
(
7
)
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2
⋅
3
2
x
7
=
30
2 \cdot 3^{\frac{2 x}{7}}=30
2
⋅
3
7
2
x
=
30
\newline
Which of the following is the solution of the equation?
\newline
Choose
1
1
1
answer:
\newline
(A)
x
=
7
2
log
3
(
15
)
x=\frac{7}{2} \log _{3}(15)
x
=
2
7
lo
g
3
(
15
)
\newline
(B)
x
=
7
2
log
15
(
3
)
x=\frac{7}{2} \log _{15}(3)
x
=
2
7
lo
g
15
(
3
)
\newline
C)
x
=
7
2
log
30
(
6
)
x=\frac{7}{2} \log _{30}(6)
x
=
2
7
lo
g
30
(
6
)
\newline
(D)
x
=
7
2
log
6
(
30
)
x=\frac{7}{2} \log _{6}(30)
x
=
2
7
lo
g
6
(
30
)
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Which of the following is equivalent to
1
log
3
(
m
)
\frac{1}{\log _{3}(m)}
l
o
g
3
(
m
)
1
?
\newline
Choose
1
1
1
answer:
\newline
(A)
log
m
(
3
)
\log _{m}(3)
lo
g
m
(
3
)
\newline
(B)
log
3
(
m
)
\log _{3}(m)
lo
g
3
(
m
)
\newline
(C)
−
log
m
(
3
)
-\log _{m}(3)
−
lo
g
m
(
3
)
\newline
(D)
−
log
3
(
m
)
-\log _{3}(m)
−
lo
g
3
(
m
)
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Which of the following is equivalent to
log
(
3
)
log
n
(
3
)
\frac{\log (3)}{\log _{n}(3)}
l
o
g
n
(
3
)
l
o
g
(
3
)
?
\newline
Choose
1
1
1
answer:
\newline
(A)
log
(
n
)
\log (n)
lo
g
(
n
)
\newline
(B)
log
3
(
n
)
\log _{3}(n)
lo
g
3
(
n
)
\newline
(C)
1
log
(
n
)
\frac{1}{\log (n)}
l
o
g
(
n
)
1
\newline
(D)
1
log
3
(
n
)
\frac{1}{\log _{3}(n)}
l
o
g
3
(
n
)
1
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Which of the following is equivalent to
1
log
b
(
4
)
\frac{1}{\log _{b}(4)}
l
o
g
b
(
4
)
1
?
\newline
Choose
1
1
1
answer:
\newline
(A)
log
b
(
4
)
\log _{b}(4)
lo
g
b
(
4
)
\newline
(B)
log
4
(
b
)
\log _{4}(b)
lo
g
4
(
b
)
\newline
(C)
−
log
b
(
4
)
-\log _{b}(4)
−
lo
g
b
(
4
)
\newline
(D)
−
log
4
(
b
)
-\log _{4}(b)
−
lo
g
4
(
b
)
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Which of the following is equivalent to
log
3
(
a
)
⋅
log
(
3
)
\log _{3}(a) \cdot \log (3)
lo
g
3
(
a
)
⋅
lo
g
(
3
)
?
\newline
Choose
1
1
1
answer:
\newline
(A)
log
(
a
)
\log (a)
lo
g
(
a
)
\newline
(B)
log
(
3
)
\log (3)
lo
g
(
3
)
\newline
(C)
log
(
3
a
)
\log (3 a)
lo
g
(
3
a
)
\newline
(D)
log
3
(
3
a
)
\log _{3}(3 a)
lo
g
3
(
3
a
)
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Which of the following is equivalent to
log
5
(
m
)
log
15
(
m
)
\frac{\log _{5}(m)}{\log _{15}(m)}
l
o
g
15
(
m
)
l
o
g
5
(
m
)
?
\newline
Choose
1
1
1
answer:
\newline
(A)
1
3
\frac{1}{3}
3
1
\newline
(B)
3
3
3
\newline
(C)
log
(
3
)
\log (3)
lo
g
(
3
)
\newline
(D)
log
5
(
15
)
\log _{5}(15)
lo
g
5
(
15
)
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Which of the following is equivalent to
log
c
(
6
)
log
(
6
)
\frac{\log _{c}(6)}{\log (6)}
l
o
g
(
6
)
l
o
g
c
(
6
)
?
\newline
Choose
1
1
1
answer:
\newline
(A)
log
(
c
)
\log (c)
lo
g
(
c
)
\newline
(B)
log
c
(
1
)
\log _{c}(1)
lo
g
c
(
1
)
\newline
(C)
1
log
(
c
)
\frac{1}{\log (c)}
l
o
g
(
c
)
1
\newline
(D)
1
log
(
6
)
\frac{1}{\log (6)}
l
o
g
(
6
)
1
Get tutor help
Evaluate the logarithm.
\newline
Round your answer to the nearest thousandth.
\newline
log
4
(
1
19
)
≈
\log _{4}\left(\frac{1}{19}\right) \approx
lo
g
4
(
19
1
)
≈
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Evaluate the logarithm.
\newline
Round your answer to the nearest thousandth.
\newline
log
2
(
1
50
)
≈
\log _{2}\left(\frac{1}{50}\right) \approx
lo
g
2
(
50
1
)
≈
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4
⋅
5
−
6
t
=
2000
4 \cdot 5^{-6 t}=2000
4
⋅
5
−
6
t
=
2000
\newline
Which of the following is the solution of the equation?
\newline
Choose
1
1
1
answer:
\newline
(A)
t
=
−
log
5
(
500
)
6
t=\frac{-\log _{5}(500)}{6}
t
=
6
−
l
o
g
5
(
500
)
\newline
(B)
t
=
−
log
20
(
2000
)
6
t=\frac{-\log _{20}(2000)}{6}
t
=
6
−
l
o
g
20
(
2000
)
\newline
(C)
t
=
−
log
2000
(
20
)
6
t=\frac{-\log _{2000}(20)}{6}
t
=
6
−
l
o
g
2000
(
20
)
\newline
(D)
t
=
−
log
500
(
5
)
6
t=\frac{-\log _{500}(5)}{6}
t
=
6
−
l
o
g
500
(
5
)
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Which of the following is equivalent to
log
2
(
c
)
⋅
log
c
(
2
)
\log _{2}(c) \cdot \log _{c}(2)
lo
g
2
(
c
)
⋅
lo
g
c
(
2
)
?
\newline
Choose
1
1
1
answer:
\newline
(A)
log
(
c
)
\log (c)
lo
g
(
c
)
\newline
(B)
log
(
2
)
\log (2)
lo
g
(
2
)
\newline
(C)
1
1
1
\newline
(D)
−
1
-1
−
1
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Which of the following is equivalent to
log
c
(
16
)
⋅
log
2
(
c
)
\log _{c}(16) \cdot \log _{2}(c)
lo
g
c
(
16
)
⋅
lo
g
2
(
c
)
?
\newline
Choose
1
1
1
answer:
\newline
(A)
4
4
4
\newline
(B)
8
8
8
\newline
(C)
log
(
8
)
\log (8)
lo
g
(
8
)
\newline
(D)
log
c
(
4
)
\log _{c}(4)
lo
g
c
(
4
)
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Which of the following is equivalent to
log
3
(
b
)
⋅
log
b
(
27
)
\log _{3}(b) \cdot \log _{b}(27)
lo
g
3
(
b
)
⋅
lo
g
b
(
27
)
?
\newline
Choose
1
1
1
answer:
\newline
(A)
3
3
3
\newline
(B)
9
9
9
\newline
(C)
log
(
9
)
\log (9)
lo
g
(
9
)
\newline
(D)
log
b
(
3
)
\log _{b}(3)
lo
g
b
(
3
)
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Which of the following is equivalent to
log
4
(
m
)
⋅
log
m
(
20
)
\log _{4}(m) \cdot \log _{m}(20)
lo
g
4
(
m
)
⋅
lo
g
m
(
20
)
?
\newline
Choose
1
1
1
answer:
\newline
(A)
5
5
5
\newline
(B)
80
80
80
\newline
(C)
log
(
5
)
\log (5)
lo
g
(
5
)
\newline
(D)
log
4
(
20
)
\log _{4}(20)
lo
g
4
(
20
)
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Which of the following is equivalent to
log
9
(
m
)
log
(
m
)
\frac{\log _{9}(m)}{\log (m)}
l
o
g
(
m
)
l
o
g
9
(
m
)
?
\newline
Choose
1
1
1
answer:
\newline
(A)
log
(
9
)
\log (9)
lo
g
(
9
)
\newline
(B)
log
9
(
1
)
\log _{9}(1)
lo
g
9
(
1
)
\newline
(C)
1
log
(
m
)
\frac{1}{\log (m)}
l
o
g
(
m
)
1
\newline
(D)
1
log
(
9
)
\frac{1}{\log (9)}
l
o
g
(
9
)
1
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Which of the following is equivalent to
log
(
a
)
⋅
log
a
(
5
)
\log (a) \cdot \log _{a}(5)
lo
g
(
a
)
⋅
lo
g
a
(
5
)
?
\newline
Choose
1
1
1
answer:
\newline
(A)
log
(
a
)
\log (a)
lo
g
(
a
)
\newline
(B)
log
(
5
)
\log (5)
lo
g
(
5
)
\newline
(C)
log
(
5
a
)
\log (5 a)
lo
g
(
5
a
)
\newline
(D)
log
a
(
5
a
)
\log _{a}(5 a)
lo
g
a
(
5
a
)
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Which of the following is equivalent to
log
7
(
a
)
⋅
log
b
(
7
)
\log _{7}(a) \cdot \log _{b}(7)
lo
g
7
(
a
)
⋅
lo
g
b
(
7
)
?
\newline
Choose
1
1
1
answer:
\newline
(A)
log
(
7
)
\log (7)
lo
g
(
7
)
\newline
(B)
log
(
7
a
)
\log (7 a)
lo
g
(
7
a
)
\newline
(C)
log
a
(
b
)
\log _{a}(b)
lo
g
a
(
b
)
\newline
(D)
log
b
(
a
)
\log _{b}(a)
lo
g
b
(
a
)
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Which of the following is equivalent to
log
a
(
18
)
⋅
log
3
(
a
)
\log _{a}(18) \cdot \log _{3}(a)
lo
g
a
(
18
)
⋅
lo
g
3
(
a
)
?
\newline
Choose
1
1
1
answer:
\newline
(A)
6
6
6
\newline
(B)
log
(
6
)
\log (6)
lo
g
(
6
)
\newline
(C)
log
3
(
18
)
\log _{3}(18)
lo
g
3
(
18
)
\newline
(D)
log
18
(
3
)
\log _{18}(3)
lo
g
18
(
3
)
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Which of the following is equivalent to
log
(
t
)
log
8
(
t
)
\frac{\log (t)}{\log _{8}(t)}
l
o
g
8
(
t
)
l
o
g
(
t
)
?
\newline
Choose
1
1
1
answer:
\newline
(A)
log
(
8
)
\log (8)
lo
g
(
8
)
\newline
(B)
log
8
(
t
2
)
\log _{8}\left(t^{2}\right)
lo
g
8
(
t
2
)
\newline
(C)
1
log
(
8
)
\frac{1}{\log (8)}
l
o
g
(
8
)
1
\newline
(D)
1
log
8
(
t
2
)
\frac{1}{\log _{8}\left(t^{2}\right)}
l
o
g
8
(
t
2
)
1
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Which of the following is equivalent to
log
b
(
8
)
log
b
(
2
)
\frac{\log _{b}(8)}{\log _{b}(2)}
l
o
g
b
(
2
)
l
o
g
b
(
8
)
?
\newline
Choose
1
1
1
answer:
\newline
(A)
3
3
3
\newline
(B)
4
4
4
\newline
(C)
log
b
(
3
)
\log _{b}(3)
lo
g
b
(
3
)
\newline
(D)
log
(
4
)
\log (4)
lo
g
(
4
)
Get tutor help
Expand the logarithm. Assume all expressions exist and are well-defined.
\newline
Write your answer as a sum or difference of common logarithms or multiples of common logarithms. The inside of each logarithm must be a distinct constant or variable.
\newline
log
(
u
v
)
\log(uv)
lo
g
(
uv
)
\newline
_____
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