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MAA AMC
AMC 12/AHSME
1964 AMC 12/AHSME
21
Logs
Logs
Source: 1964 AHSME Problem 21
August 23, 2013
logarithms
quadratics
AMC
Problem Statement
If
log
b
2
x
+
log
x
2
b
=
1
,
b
>
0
,
b
≠
1
,
x
≠
1
\log_{b^2}x+\log_{x^2}b=1, b>0, b \neq 1, x \neq 1
lo
g
b
2
x
+
lo
g
x
2
b
=
1
,
b
>
0
,
b
=
1
,
x
=
1
, then
x
x
x
equals:
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
1
/
b
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
1
/
b
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
b
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
b
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
b
<span class='latex-bold'>(A)</span>\ 1/b^2 \qquad<span class='latex-bold'>(B)</span>\ 1/b \qquad<span class='latex-bold'>(C)</span>\ b^2 \qquad<span class='latex-bold'>(D)</span>\ b \qquad<span class='latex-bold'>(E)</span>\ \sqrt{b}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
1/
b
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
1/
b
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
b
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
b
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
b
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