How to simplify this?
$\displaystyle\frac{n^{\log m}}{m^{\log n}}$
How to simplify this?
$\displaystyle\frac{n^{\log m}}{m^{\log n}}$
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 $m^{\log n} = e^{\log (m^{\log n})} = e^{\log n \log m}$
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 $n^{\log m} = m^{\log n}$.