Definition

The arithmetic function $$f_{b}(n)=\left[ \frac{\log(n)}{\log(b)} \right]+1$$where, $[x]$ is the floor function. This function takes a natural number and tells us the number of digits of $n$ in its expansion in base $b$.

Example

$$f_{2}(10)=4$$ since the binary expansion of $10$ is $1010$.