MathDB
Problems
Contests
National and Regional Contests
USA Contests
MAA AMC
AMC 12/AHSME
1976 AMC 12/AHSME
20
20
Part of
1976 AMC 12/AHSME
Problems
(1)
a,b,x satisfy logarithmic equation
Source: 1976 AHSME Problem 20
5/18/2014
Let
a
,
b
,
a,~b,
a
,
b
,
and
x
x
x
be positive real numbers distinct from one. Then
4
(
log
a
x
)
2
+
3
(
log
b
x
)
2
=
8
(
log
a
x
)
(
log
b
x
)
4(\log_ax)^2+3(\log_bx)^2=8(\log_ax)(\log_bx)
4
(
lo
g
a
x
)
2
+
3
(
lo
g
b
x
)
2
=
8
(
lo
g
a
x
)
(
lo
g
b
x
)
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
for all values of
a
,
b
,
and
x
<span class='latex-bold'>(A) </span>\text{for all values of }a,~b,\text{ and }x\qquad
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
for all values of
a
,
b
,
and
x
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
if and only if
a
=
b
2
<span class='latex-bold'>(B) </span>\text{if and only if }a=b^2\qquad
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
if and only if
a
=
b
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
if and only if
b
=
a
2
<span class='latex-bold'>(C) </span>\text{if and only if }b=a^2\qquad
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
if and only if
b
=
a
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
if and only if
x
=
a
b
<span class='latex-bold'>(D) </span>\text{if and only if }x=ab\qquad
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
if and only if
x
=
ab
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
for none of these
<span class='latex-bold'>(E) </span>\text{for none of these}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
for none of these
logarithms
quadratics
AMC