Base conversion is the problem of converting an integer between representations in two fixed bases. Without loss of generality consider the case of relatively prime bases. I think it's easier to imagine a single type of machine that can handle all cases even though the bases are fixed, so consider a base-$b$ number as being represented by a concatenation of $\lceil \log_2{(b)} \rceil$-bit words, and let our multi-tape Turing machines have a binary input and output alphabet. But equivalently we can build a machine whose input and output tape alphabet sizes match the source and target bases.
Is base conversion known to be computable in better than essentially quadratic time?
This question has been asked many times but every reference I can find is in a practical context. I have seen claims that it can be accomplished in quasilinear time, but I can't understand how this is supposed to work, in part because assumptions about the model are often unstated in such discussions.