Simplifying this expression

Question

How to simplify this?

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

Answer 1

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 $m^{\log n} = e^{\log (m^{\log n})} = e^{\log n \log m}$

Answer 2

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.

Answer 3

Note that $n^{\log m} = m^{\log n}$.