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Math Problems
Precalculus
Quotient property of logarithms
Cho biểu thức
\newline
P
=
1
log
a
2
2
+
1
log
a
4
2
+
…
.
.
+
1
log
a
2
n
2
,
P=\frac{1}{\log_{a^{2}}2}+\frac{1}{\log_{a^{4}}2}+\dots..+\frac{1}{\log_{a^{2n}}2},
P
=
lo
g
a
2
2
1
+
lo
g
a
4
2
1
+
…
..
+
lo
g
a
2
n
2
1
,
với
\newline
a
a
a
l\`a số thực lớn hơn
\newline
1
,
n
∈
N
∗
∗
.
1,n \in \mathbb{N}^{**}.
1
,
n
∈
N
∗∗
.
Biết
\newline
P
≤
100
log
2
a
,
P \leq 100\log_{2}a,
P
≤
100
lo
g
2
a
,
t\'inh tổng các giá trị của
\newline
n
.
n.
n
.
\newline
A.
9
9
9
.
\newline
B.
100
100
100
.
\newline
C.
0
0
0
.
\newline
D.
45
45
45
.
Get tutor help
log
3
(
−
cos
x
)
−
log
9
(
sin
x
)
+
1
4
=
−
log
9
2
\log _{3}(-\cos x)-\log _{9}(\sin x)+\frac{1}{4}=-\log _{9} 2
lo
g
3
(
−
cos
x
)
−
lo
g
9
(
sin
x
)
+
4
1
=
−
lo
g
9
2
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The function
f
f
f
is given by
f
(
x
)
=
log
2
x
f(x)=\log _{2} x
f
(
x
)
=
lo
g
2
x
. What input value in the domain of
f
f
f
yields an output value of
4
4
4
?
\newline
(A)
32
32
32
\newline
f
(
x
)
=
log
2
x
f(x)=\log _{2} x
f
(
x
)
=
lo
g
2
x
\newline
(B)
16
16
16
\newline
(C)
2
2
2
\newline
log
2
8
=
3
\log _{2} 8=3
lo
g
2
8
=
3
\newline
(D)
1
2
\frac{1}{2}
2
1
\newline
log
2
x
=
4
2
3
=
8
\log _{2} x=4 \quad 2^{3}=8
lo
g
2
x
=
4
2
3
=
8
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log
8
1
64
=
\log _{8} \frac{1}{64}=
lo
g
8
64
1
=
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27
3
=
5
x
3
\frac{27}{3}=\frac{5 x}{3}
3
27
=
3
5
x
\newline
Watch on
\newline
9
=
9=
9
=
\newline
Youtulube
\newline
(a) If
log
5
x
=
9
\log _{5} x=9
lo
g
5
x
=
9
, then
x
=
x=
x
=
□
\square
□
\newline
(b) If
log
7
x
=
4
\log _{7} x=4
lo
g
7
x
=
4
, then
x
=
x=
x
=
□
\square
□
\newline
Submit Question
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x
=
17
log
2
(
2
x
)
+
log
2
(
x
−
7
)
=
log
2
(
4
x
)
\begin{array}{c}x=17 \\ \log _{2}(2 x)+\log _{2}(x-7)=\log _{2}(4 x)\end{array}
x
=
17
lo
g
2
(
2
x
)
+
lo
g
2
(
x
−
7
)
=
lo
g
2
(
4
x
)
Get tutor help
11
11
11
)
d
d
x
(
log
10
x
+
log
x
10
+
log
x
x
+
log
10
10
)
\frac{d}{d x}\left(\log _{10} x+\log _{x} 10+\log _{x} x+\log _{10} 10\right)
d
x
d
(
lo
g
10
x
+
lo
g
x
10
+
lo
g
x
x
+
lo
g
10
10
)
Get tutor help
6
6
6
\newline
4
4
4
(a) Given
log
2
3
=
p
\log _{2} 3=p
lo
g
2
3
=
p
and
log
2
5
=
q
\log _{2} 5=q
lo
g
2
5
=
q
, express and simplify
log
8
2
5
+
log
3
2
\log _{8} \frac{2}{5}+\log _{3} 2
lo
g
8
5
2
+
lo
g
3
2
in terms of
p
p
p
and
q
q
q
.
\newline
[
3
3
3
]
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log
7
64
−
log
7
8
3
+
log
7
2
=
log
7
4
p
\log _{7} 64-\log _{7} \frac{8}{3}+\log _{7} 2=\log _{7} 4 p
lo
g
7
64
−
lo
g
7
3
8
+
lo
g
7
2
=
lo
g
7
4
p
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log
(
x
+
2
)
+
log
(
x
)
−
1
=
log
(
3
2
)
\log(x+2)+\log(x)-1=\log\left(\frac{3}{2}\right)
lo
g
(
x
+
2
)
+
lo
g
(
x
)
−
1
=
lo
g
(
2
3
)
Get tutor help
Question
3
3
3
\newline
What is the solution to the equati
\newline
log
6
4
x
2
−
log
6
x
=
2
\log _{6} 4 x^{2}-\log _{6} x=2
lo
g
6
4
x
2
−
lo
g
6
x
=
2
\newline
A.
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Evaluate the logarithmic expression without using a calculator. Remember that
log
a
x
\log _{a} x
lo
g
a
x
is the exponent to which a must be raised in order to obtain
x
\mathrm{x}
x
.
\newline
(a)
log
2
16
\log _{2} 16
lo
g
2
16
\newline
(d)
log
2
2
\log _{2} \sqrt{2}
lo
g
2
2
\newline
(b)
log
3
1
\log _{3} 1
lo
g
3
1
\newline
(e)
log
e
(
1
e
2
)
\log _{e}\left(\frac{1}{e^{2}}\right)
lo
g
e
(
e
2
1
)
\newline
(c)
log
10
0.1
\log _{10} 0.1
lo
g
10
0.1
\newline
(f)
log
1
/
2
8
\log _{1 / 2} 8
lo
g
1/2
8
.
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Use the values
log
48
≈
1.68
\log 48 \approx 1.68
lo
g
48
≈
1.68
and
log
3
≈
0.48
\log 3 \approx 0.48
lo
g
3
≈
0.48
to find the approximate value of
log
3
48
\log _{3} 48
lo
g
3
48
.
\newline
log
3
48
≈
\log _{3} 48 \approx
lo
g
3
48
≈
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2
2
2
. Find the value of
x
x
x
and
y
y
y
which satisfy the equations
\newline
9
x
3
y
=
1
9
log
2
x
=
8
1
3
−
log
2
y
\begin{array}{c} \frac{9^{x}}{3^{y}}=\frac{1}{9} \\ \log _{2} x=8^{\frac{1}{3}}-\log _{2} y \end{array}
3
y
9
x
=
9
1
lo
g
2
x
=
8
3
1
−
lo
g
2
y
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Solve for a positive value of
x
x
x
.
\newline
log
x
(
36
)
=
2
\log _{x}(36)=2
lo
g
x
(
36
)
=
2
\newline
Answer:
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Solve
(
log
81
x
)
(
1
log
x
3
)
=
6
1
4
\left(\log _{81} x\right)\left(\frac{1}{\log _{x} 3}\right)=6 \frac{1}{4}
(
lo
g
81
x
)
(
l
o
g
x
3
1
)
=
6
4
1
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15
15
15
The figure shows a large square of side
84
c
m
.
A
,
B
84 \mathrm{~cm} . \mathrm{A}, \mathrm{B}
84
cm
.
A
,
B
and
C
\mathrm{C}
C
are mid-points of three sides of the square. The large square is made up of four smaller identical squares. Inside each of the smaller squares is a semicircle and a quarter circle. Find the total area of the shaded parts. (Take
π
=
22
7
\pi=\frac{22}{7}
π
=
7
22
)
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(iv)
f
(
x
)
=
(
x
+
1
2
)
log
(
x
+
1
2
)
{
x
2
+
2
x
−
3
4
x
2
−
4
x
−
3
}
f(x)=\left(x+\frac{1}{2}\right) \log _{\left(x+\frac{1}{2}\right)}\left\{\frac{x^{2}+2 x-3}{4 x^{2}-4 x-3}\right\}
f
(
x
)
=
(
x
+
2
1
)
lo
g
(
x
+
2
1
)
{
4
x
2
−
4
x
−
3
x
2
+
2
x
−
3
}
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5
5
5
.
log
3
x
−
4
log
x
3
=
1
+
1
log
9
3
\log _{3} x-4 \log _{x} 3=1+\frac{1}{\log _{9} 3}
lo
g
3
x
−
4
lo
g
x
3
=
1
+
l
o
g
9
3
1
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If
log
(
x
+
1
)
64
=
3
\log _{(x+1)} 64=3
lo
g
(
x
+
1
)
64
=
3
, find the value of
Get tutor help
Expand the logarithm fully using the properties of logs. Express the final answer in terms of
log
x
,
log
y
\log x, \log y
lo
g
x
,
lo
g
y
, and
log
z
\log z
lo
g
z
.
\newline
log
y
x
4
z
4
3
\log \frac{y x^{4}}{\sqrt[3]{z^{4}}}
lo
g
3
z
4
y
x
4
\newline
Answer:
Get tutor help
Expand the logarithm fully using the properties of logs. Express the final answer in terms of
log
x
,
log
y
\log x, \log y
lo
g
x
,
lo
g
y
, and
log
z
\log z
lo
g
z
.
\newline
log
z
4
3
y
4
x
5
\log \frac{\sqrt[3]{z^{4}}}{y^{4} x^{5}}
lo
g
y
4
x
5
3
z
4
\newline
Answer:
Get tutor help
Expand the logarithm fully using the properties of logs. Express the final answer in terms of
log
x
,
log
y
\log x, \log y
lo
g
x
,
lo
g
y
, and
log
z
\log z
lo
g
z
.
\newline
log
x
4
y
3
z
5
\log \frac{x^{4} \sqrt{y^{3}}}{z^{5}}
lo
g
z
5
x
4
y
3
\newline
Answer:
Get tutor help
Expand the logarithm fully using the properties of logs. Express the final answer in terms of
log
x
,
log
y
\log x, \log y
lo
g
x
,
lo
g
y
, and
log
z
\log z
lo
g
z
.
\newline
log
z
4
3
y
x
2
\log \frac{\sqrt[3]{z^{4}} y}{x^{2}}
lo
g
x
2
3
z
4
y
\newline
Answer:
Get tutor help
Expand the logarithm fully using the properties of logs. Express the final answer in terms of
log
x
,
log
y
\log x, \log y
lo
g
x
,
lo
g
y
, and
log
z
\log z
lo
g
z
.
\newline
log
z
x
3
y
3
\log \frac{z x^{3}}{\sqrt[3]{y}}
lo
g
3
y
z
x
3
\newline
Answer:
Get tutor help
Expand the logarithm fully using the properties of logs. Express the final answer in terms of
log
x
,
log
y
\log x, \log y
lo
g
x
,
lo
g
y
, and
log
z
\log z
lo
g
z
.
\newline
log
z
x
2
y
4
\log \frac{z x^{2}}{y^{4}}
lo
g
y
4
z
x
2
\newline
Answer:
Get tutor help
Expand the logarithm fully using the properties of logs. Express the final answer in terms of
log
x
,
log
y
\log x, \log y
lo
g
x
,
lo
g
y
, and
log
z
\log z
lo
g
z
.
\newline
log
x
3
z
y
2
\log \frac{\sqrt[3]{x}}{z y^{2}}
lo
g
z
y
2
3
x
\newline
Answer:
Get tutor help
Expand the logarithm fully using the properties of logs. Express the final answer in terms of
log
x
,
log
y
\log x, \log y
lo
g
x
,
lo
g
y
, and
log
z
\log z
lo
g
z
.
\newline
log
x
2
z
y
2
\log \frac{x^{2}}{\sqrt{z} y^{2}}
lo
g
z
y
2
x
2
\newline
Answer:
Get tutor help
Expand the logarithm fully using the properties of logs. Express the final answer in terms of
log
x
,
log
y
\log x, \log y
lo
g
x
,
lo
g
y
, and
log
z
\log z
lo
g
z
.
\newline
log
y
z
4
x
2
\log \frac{\sqrt{y} z^{4}}{x^{2}}
lo
g
x
2
y
z
4
\newline
Answer:
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Find the numerical value of the log expression.
\newline
log
a
=
−
10
log
b
=
10
log
c
=
11
log
a
b
5
c
9
\begin{array}{cc} \log a=-10 \quad & \log b=10 \quad \log c=11 \\ & \log \frac{\sqrt{a b^{5}}}{c^{9}} \end{array}
lo
g
a
=
−
10
lo
g
b
=
10
lo
g
c
=
11
lo
g
c
9
a
b
5
\newline
Answer:
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Find the numerical value of the log expression.
\newline
log
a
=
7
log
b
=
4
log
c
=
6
log
b
8
a
6
c
5
\begin{array}{c} \log a=7 \quad \log b=4 \quad \log c=6 \\ \log \frac{b^{8}}{a^{6} c^{5}} \end{array}
lo
g
a
=
7
lo
g
b
=
4
lo
g
c
=
6
lo
g
a
6
c
5
b
8
\newline
Answer:
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Find the numerical value of the log expression.
\newline
log
a
=
4
log
b
=
−
8
log
c
=
−
3
log
a
3
b
5
c
5
\begin{array}{c} \log a=4 \quad \log b=-8 \quad \log c=-3 \\ \log \frac{\sqrt{a^{3} b^{5}}}{c^{5}} \end{array}
lo
g
a
=
4
lo
g
b
=
−
8
lo
g
c
=
−
3
lo
g
c
5
a
3
b
5
\newline
Answer:
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Find the numerical value of the log expression.
\newline
log
a
=
9
log
b
=
6
log
c
=
4
log
b
5
c
5
a
9
\begin{array}{c} \log a=9 \quad \log b=6 \quad \log c=4 \\ \log \frac{\sqrt{b^{5} c^{5}}}{a^{9}} \end{array}
lo
g
a
=
9
lo
g
b
=
6
lo
g
c
=
4
lo
g
a
9
b
5
c
5
\newline
Answer:
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Find the numerical value of the log expression.
\newline
log
a
=
3
log
b
=
9
log
c
=
−
2
log
a
5
b
5
3
c
6
\begin{array}{c} \log a=3 \quad \log b=9 \quad \log c=-2 \\ \log \frac{\sqrt[3]{a^{5} b^{5}}}{c^{6}} \end{array}
lo
g
a
=
3
lo
g
b
=
9
lo
g
c
=
−
2
lo
g
c
6
3
a
5
b
5
\newline
Answer:
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Find the numerical value of the log expression.
\newline
log
a
=
2
log
b
=
5
log
c
=
−
8
log
b
5
a
3
c
5
\begin{array}{c} \log a=2 \quad \log b=5 \quad \log c=-8 \\ \log \frac{b^{5}}{\sqrt{a^{3} c^{5}}} \end{array}
lo
g
a
=
2
lo
g
b
=
5
lo
g
c
=
−
8
lo
g
a
3
c
5
b
5
\newline
Answer:
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Find the numerical value of the log expression.
\newline
log
a
=
8
log
b
=
1
log
c
=
−
4
log
a
5
b
4
c
3
\begin{array}{c} \log a=8 \quad \log b=1 \quad \log c=-4 \\ \log \frac{a^{5} b^{4}}{\sqrt{c^{3}}} \end{array}
lo
g
a
=
8
lo
g
b
=
1
lo
g
c
=
−
4
lo
g
c
3
a
5
b
4
\newline
Answer:
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Find the numerical value of the log expression.
\newline
log
a
=
1
log
b
=
6
log
c
=
9
log
a
4
b
7
c
4
3
\begin{array}{cc} \log a=1 \quad & \log b=6 \quad \log c=9 \\ \log \frac{a^{4}}{\sqrt[3]{b^{7} c^{4}}} \end{array}
lo
g
a
=
1
lo
g
3
b
7
c
4
a
4
lo
g
b
=
6
lo
g
c
=
9
\newline
Answer:
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Find the numerical value of the log expression.
\newline
log
a
=
10
log
b
=
7
log
c
=
−
6
log
c
a
4
b
5
\begin{array}{c} \log a=10 \quad \log b=7 \quad \log c=-6 \\ \log \frac{\sqrt{c}}{a^{4} b^{5}} \end{array}
lo
g
a
=
10
lo
g
b
=
7
lo
g
c
=
−
6
lo
g
a
4
b
5
c
\newline
Answer:
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Find the numerical value of the log expression.
\newline
log
a
=
−
5
log
b
=
8
log
c
=
3
log
c
3
a
9
b
8
\begin{array}{c} \log a=-5 \quad \log b=8 \quad \log c=3 \\ \log \frac{c^{3}}{a^{9} b^{8}} \end{array}
lo
g
a
=
−
5
lo
g
b
=
8
lo
g
c
=
3
lo
g
a
9
b
8
c
3
\newline
Answer:
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Find the numerical value of the log expression.
\newline
log
a
=
12
log
b
=
12
log
c
=
−
6
log
a
2
b
9
c
5
\begin{array}{c} \log a=12 \quad \log b=12 \quad \log c=-6 \\ \log \frac{a^{2}}{b^{9} c^{5}} \end{array}
lo
g
a
=
12
lo
g
b
=
12
lo
g
c
=
−
6
lo
g
b
9
c
5
a
2
\newline
Answer:
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Find the numerical value of the log expression.
\newline
log
a
=
11
log
b
=
−
7
log
c
=
−
12
log
c
3
a
9
b
6
\begin{array}{c} \log a=11 \quad \log b=-7 \quad \log c=-12 \\ \log \frac{\sqrt[3]{c}}{a^{9} b^{6}} \end{array}
lo
g
a
=
11
lo
g
b
=
−
7
lo
g
c
=
−
12
lo
g
a
9
b
6
3
c
\newline
Answer:
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Find the numerical value of the log expression.
\newline
log
a
=
9
log
b
=
2
log
c
=
−
6
log
c
5
3
a
9
b
4
\begin{array}{c} \log a=9 \quad \log b=2 \quad \log c=-6 \\ \log \frac{\sqrt[3]{c^{5}}}{a^{9} b^{4}} \end{array}
lo
g
a
=
9
lo
g
b
=
2
lo
g
c
=
−
6
lo
g
a
9
b
4
3
c
5
\newline
Answer:
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Find the numerical value of the log expression.
\newline
log
a
=
−
10
log
b
=
11
log
c
=
4
log
c
4
a
3
b
3
\begin{array}{c} \log a=-10 \quad \log b=11 \quad \log c=4 \\ \log \frac{c^{4}}{a^{3} b^{3}} \end{array}
lo
g
a
=
−
10
lo
g
b
=
11
lo
g
c
=
4
lo
g
a
3
b
3
c
4
\newline
Answer:
Get tutor help
Find the numerical value of the log expression.
\newline
log
a
=
8
log
b
=
2
log
c
=
−
4
log
c
8
a
9
b
9
\begin{array}{c} \log a=8 \quad \log b=2 \quad \log c=-4 \\ \log \frac{c^{8}}{a^{9} b^{9}} \end{array}
lo
g
a
=
8
lo
g
b
=
2
lo
g
c
=
−
4
lo
g
a
9
b
9
c
8
\newline
Answer:
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Write the expression below as a single logarithm in simplest form.
\newline
2
log
b
3
−
log
b
3
2 \log _{b} 3-\log _{b} 3
2
lo
g
b
3
−
lo
g
b
3
\newline
Answer:
log
b
(
□
)
\log _{b}(\square)
lo
g
b
(
□
)
Get tutor help
Find the numerical value of the log expression.
\newline
log
a
=
2
log
b
=
2
log
c
=
2
log
c
7
a
9
b
5
\begin{array}{c} \log a=2 \quad \log b=2 \quad \log c=2 \\ \log \frac{c^{7}}{a^{9} b^{5}} \end{array}
lo
g
a
=
2
lo
g
b
=
2
lo
g
c
=
2
lo
g
a
9
b
5
c
7
\newline
Answer:
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Find the numerical value of the log expression.
\newline
log
a
=
−
3
log
b
=
−
11
log
c
=
8
log
a
3
b
4
c
5
\begin{array}{c} \log a=-3 \quad \log b=-11 \quad \log c=8 \\ \log \frac{\sqrt[3]{a}}{b^{4} c^{5}} \end{array}
lo
g
a
=
−
3
lo
g
b
=
−
11
lo
g
c
=
8
lo
g
b
4
c
5
3
a
\newline
Answer:
Get tutor help
Find the numerical value of the log expression.
\newline
log
a
=
−
8
log
b
=
9
log
c
=
−
3
log
a
9
b
9
c
8
\begin{array}{c} \log a=-8 \quad \log b=9 \quad \log c=-3 \\ \log \frac{a^{9} b^{9}}{c^{8}} \end{array}
lo
g
a
=
−
8
lo
g
b
=
9
lo
g
c
=
−
3
lo
g
c
8
a
9
b
9
\newline
Answer:
Get tutor help
Find the numerical value of the log expression.
\newline
log
a
=
−
6
log
b
=
−
9
log
c
=
−
12
log
b
4
c
7
3
a
4
\begin{array}{c} \log a=-6 \quad \log b=-9 \quad \log c=-12 \\ \log \frac{\sqrt[3]{b^{4} c^{7}}}{a^{4}} \end{array}
lo
g
a
=
−
6
lo
g
b
=
−
9
lo
g
c
=
−
12
lo
g
a
4
3
b
4
c
7
\newline
Answer:
Get tutor help
Find the numerical value of the log expression.
\newline
log
a
=
−
6
log
b
=
10
log
c
=
7
log
b
a
5
c
7
\begin{array}{c} \log a=-6 \quad \log b=10 \quad \log c=7 \\ \log \frac{\sqrt{b}}{a^{5} c^{7}} \end{array}
lo
g
a
=
−
6
lo
g
b
=
10
lo
g
c
=
7
lo
g
a
5
c
7
b
\newline
Answer:
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