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AMC 12/AHSME
1959 AMC 12/AHSME
45
Logarithmic equation
Logarithmic equation
Source: 1959 AHSME Problem 45
August 14, 2013
logarithms
AMC
logarithm
algebra
AMC 12
Problem Statement
If
(
log
3
x
)
(
log
x
2
x
)
(
log
2
x
y
)
=
log
x
x
2
\left(\log_3 x\right)\left(\log_x 2x\right)\left( \log_{2x} y\right)=\log_{x}x^2
(
lo
g
3
x
)
(
lo
g
x
2
x
)
(
lo
g
2
x
y
)
=
lo
g
x
x
2
, then
y
y
y
equals:
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
9
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
9
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
18
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
27
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
81
<span class='latex-bold'>(A)</span>\ \frac92\qquad<span class='latex-bold'>(B)</span>\ 9\qquad<span class='latex-bold'>(C)</span>\ 18\qquad<span class='latex-bold'>(D)</span>\ 27\qquad<span class='latex-bold'>(E)</span>\ 81
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
2
9
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
9
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
18
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
27
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
81
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