Logarithms and Number of Digits (Base 10)

The number of digits in $n$ (base $10$) is $\lfloor \log_{10}(n) \rfloor + 1$. For example, $n = 1234$ has $4$ digits, and $\log_{10}(1234) \approx 3.09$.

This works because $10^{k-1} \leq n < 10^k$, so $k = \lfloor \log_{10}(n) \rfloor + 1$. You can compute the number of digits without converting to a string.

Algorithms that process digits (like summing digits or reversing a number) run in $O(\log_{10} n)$ time, which is the same as $O(\log n)$ in Big-O notation.