TITLE: $\log(n)$ is what power of $n$?
QUESTION [9 upvotes]: Sorry about asking such an elementary question, but I have been wondering about this exact definition for a while. What power of $n$ is $\log(n)$. I know that it is $n^\epsilon$ for a very small $\epsilon$, but what value is $\epsilon$ exactly?

REPLY [4 votes]: Suppose we seek such $\varepsilon$, that
$$
n^\varepsilon = \log(n).
$$
Consider $n>1$. 
Then (if $\log(n)$ is natural logarithm, to the base $e$)
$$
n^\varepsilon = e^{\varepsilon\log(n)}, \qquad \log(n) = e^{\log(\log(n))},
$$
then powers of $e$ must be equal:
$$
\varepsilon \log(n) = \log(\log(n)),
$$
$$
\varepsilon = \frac{\log(\log(n))}{\log(n)}.
$$
Examples:
$n=10$:   $\varepsilon = 0.3622156886...$;
$n=10^2$:  $\varepsilon = 0.3316228421...$;
$n=10^3$: $\varepsilon = 0.2797789811...$;
$n=10^6$: $\varepsilon = 0.1900611565...$;
$n=10^9$: $\varepsilon = 0.1462731331...$.