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Canadian Mathematical Olympiad Qualification Repechage
2010 Canadian Mathematical Olympiad Qualification Repechage
1
Logarithmic Proof
Logarithmic Proof
Source: Canadian Repêchage 2010: Problem 1
May 6, 2014
logarithms
Problem Statement
Suppose that
a
a
a
,
b
b
b
and
x
x
x
are positive real numbers. Prove that
log
a
b
x
=
log
a
x
log
b
x
log
a
x
+
log
b
x
\log_{ab} x =\dfrac{\log_a x\log_b x}{\log_ax+\log_bx}
lo
g
ab
x
=
lo
g
a
x
+
lo
g
b
x
lo
g
a
x
lo
g
b
x
.
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