What kind of recurrence relations has p < 0? By the master method, 
$T(n) = aT(\frac {n}{b})+\Theta(n^k\log^pn)$
where $p$ is real.

I know $\log^4n=\log(\log(\log(\log n)))$ but how do you calculate something like $\log^pn$ where $p&lt;0$?
As Rick Decker mentions, in this context $\log^p n = (\log n)^p$, and so $p$ can be an arbitrary real number. If you wanted to denote composition, you would use $\log^{(p)} n$ (which in other contexts signifies the $p$th derivative!). In that case, when $p$ is a negative integer, you just need to use the inverse of $\log$, namely $\exp$:
$$
\begin{align*}
\log^{(2)} x &amp;= \log\log x \\
\log^{(1)} x &amp;= \log x \\
\log^{(0)} x &amp;= x \\
\log^{(-1)} x &amp;= e^x \\
\log^{(-2)} x &amp;= e^{e^x}
\end{align*}
$$
This definition ensures that $\log^{(a)} \log^{(b)} x = \log^{(a+b)} x$ for all integers $a,b$. The natural domain of this definition is integer $p$, though it can probably be extended for fractional $p$ in such a way that the equation above still holds.