How can I simplify this expression:
$$\large \frac {2^n}{n^{nlog(2)}(log(2))^{nlog(2)}}$$
My guess is that somehow I can extract a $2^n$ factor from the denominator, but how can I do that?
Would it be a change of base for the logarithms? But would a change of base ... change the answer? E.g., say I were to show that the infinite sum of this expression converges, and the logarithms given are probably assumed to have base e. Can I just change the logarithm without changing the answer?
Sorry for the wordy and simple question, but this is a bit tricky for me. And the change of base technique has shown up a few times in past questions on advanced calculus, so I just want to be sure how to do it properly. I haven't found anything on this, other than some simple formulas for base change from various online sources. But I haven't found anything that addresses the issue of raising a logarithm to a power of a logarithm.
Thanks,
To reiterate, we have the expression $$\dfrac{2^n}{n^{n\log 2}(\log 2)^{n\log 2}}$$
which looks a bit messy.
If we assume the logarithms given have base $e$ and that $n$ is a positive integer, then we can use basic properties of exponents and logs to tidy it up some.
[reminder: I'm using the convention $\log x=\log_e x=\ln x$, not to be confused with the alternative notation $\log x=\log_{10} x$.]
There are several ways to rewrite the above expression more compactly. One might consider $$\left(\dfrac{e}{n\log 2}\right)^{n\log 2}$$ which is "prettier" and allows the root test to be applied more readily.