Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

How to simplify this?

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

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

3 Answers 3

up vote 3 down vote accepted

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

share|improve this answer

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.

share|improve this answer

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

share|improve this answer
1  
I laughed at this. –  bobobinks Jan 21 '11 at 1:36
    
Sorry I stated the wrong question. It should had been "How to prove that those expressions are equal" +1 –  Wilbert Barrera Jan 21 '11 at 1:43
1  
@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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.