# Simplifying this expression

How to simplify this?

$\displaystyle\frac{n^{\log m}}{m^{\log n}}$

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Try rewriting each base. –  bobobinks Jan 21 '11 at 1:35
possible duplicate of How $a^{\log_b x} = x^{\log_b a}$ ? –  Bill Dubuque Jan 21 '11 at 2:38

First note that $x = a^{\log_a x}$

So we find that $n^{\log m} = e^{\log (n^{\log m})} = e^{\log m \log n}$

Similarly, we find that $m^{\log n} = e^{\log (m^{\log n})} = e^{\log n \log m}$

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Hint: Apply $\log$ to the whole thing and use the quotient and powers rules for $\log$. You should get a very simple result, which you can exponentiate to find your answer.
Note that $n^{\log m} = m^{\log n}$.
@Wilbert Barrera: To prove that, you can apply $\log$ to both sides, and use the fact that $\log$ is one-to-one. –  Jonas Meyer Jan 21 '11 at 1:47