What kind of recurrence relations has p < 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.