How can I simplify $N^{\frac{e}{\sqrt{\log(N)}}}$

Question

I am working through an algorithms workbook and I have the following equation: $$N^{\frac{e}{\sqrt{\log(N)}}}$$ I know I can simplify it somehow using the properties of logs and exponents but am a little rusty on that basic math. If you could provide explanation of your process I would appreciate it.

Answer 1

Use the fact that $N=e^{\log{N}} $. Then

$$N^{\frac{e}{\sqrt{\log{N}}}} = e^{e \sqrt{\log{N}}}$$

Not much more I can think of doing here.

Answer 2

$N^{\frac{e}{\sqrt{\log(N)}}}$

Take the log. $\log N^{\frac{e}{\sqrt{\log(N)}}} = \frac{e}{\sqrt{\log(N)}}\log(N) = e \sqrt{log(N)}$. Raise back to the exponent to get $N^{\frac{e}{\sqrt{\log(N)}}} = e^{ e \sqrt{log(N)}} $ which isn't much better but it has only a single instance of a variable. If we need to solve for N it will be easy.