Given a real number as an input, and an exponent, what would be the right way to determine exactly how many digits of that number are necessary to retrieve the integer part of the resultant exponent. That is, given $$$r \in \mathbb{R}$$$ and $$$n \in \mathbb{N}$$$, you'd like to find the minimum number of digits necessary to accurately compute $$$\left \lfloor{r^n}\right \rfloor$$$. Binary search of digit usage looks like it would work, but feels very inefficient, and for non-truncated real numbers (e.g. where we have a formula instead of raw digits) requires a guaranteed accurate starting point. I've studied very little numerical analysis, but I believe this falls into that realm of problems.