Are logarithmic Big-O complexities defined with constant base equal to those defined with variable base?
Let's look at the root of question: we have  $\log_2 n=\frac{\ln n}{\ln 2}$ and $\log_t n=\frac{\ln n}{\ln t}$, so $\log_2 n = \frac{\ln t}{\ln 2} \log_t n$ and when $t$ is constant, then logarithms are in same complexity class.
Bit if $t=t(n)$, then we can ruin this relation and create any class $O(f(n))$ of complexity taking $t(n)=e^{\frac{\ln n}{f(n)}}$ we have $\log_t n=\frac{\ln n}{\ln t}=f(n)$.