Meaning of $\log^{O(1)}(a+b)$
By definition
$$O(g)=\{f\colon \exists C&gt;0, \exists N \in \mathbb{N}, \forall n&gt; N, f(n)\leqslant C g(n)\}$$
so $O(1)$ can be viewed as set of bounded functions. Outgoing from here $\log^{O(1)}(a+b)$ we can understand as set of functions $\log^{f(n)}(a+b)$ where $f(n) \in O(1)$.
Addition.
Understanding of $f \in \log^{O(1)}(a+b)$ not seems to be differ from understanding of $f \in O(1)$ and does not mean that &quot;O(1) can be arbitrary large constant&quot;, as is in your comments. $O(1)$ is not constant, or set of constants, but set of functions. In same way $f \in O(1)$ does not mean that $f$ is bounded by any arbitrary large constant, but there $\exists C_f&gt;0$ for which we have $f(n) \leqslant C_f$. So, $f \in \log^{O(1)}(a+b)$, mean, that $\exists g \in O(1)$, i.e. $\exists C_g&gt;0$ $g(n) \leqslant C_g$, such that $f(n)=\log^{g(n)}(a+b)$. In words: function $f$ is bounded by $\log$ having bounded function in superscript. For example, $f$ cannot be $\log^n(a+b)$, but can be $\log^{\frac{1}{n}}(a+b)$ or $\log^{\frac{1}{n}+1}(a+b)$.