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National and Regional Contests
USA Contests
MAA AMC
AMC 12/AHSME
1963 AMC 12/AHSME
5
Logarithms less than 0
Logarithms less than 0
Source: AHSME 1963 Problem 5
January 9, 2014
logarithms
AMC
Problem Statement
If
x
x
x
and
log
10
x
\log_{10} x
lo
g
10
x
are real numbers and
log
10
x
<
0
\log_{10} x<0
lo
g
10
x
<
0
, then:
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
x
<
0
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
−
1
<
x
<
1
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
0
<
x
≤
1
<span class='latex-bold'>(A)</span>\ x<0 \qquad <span class='latex-bold'>(B)</span>\ -1<x<1 \qquad <span class='latex-bold'>(C)</span>\ 0<x\le 1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
x
<
0
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
−
1
<
x
<
1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
0
<
x
≤
1
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
−
1
<
x
<
0
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
0
<
x
<
1
<span class='latex-bold'>(D)</span>\ -1<x<0 \qquad <span class='latex-bold'>(E)</span>\ 0<x<1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
−
1
<
x
<
0
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
0
<
x
<
1
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