Find the number of digits of the number $k$ in function of $r$ and $n$

Question

Let $α∈(0,1)$ be an irrational number with infinitely digits after the decimal point. Let $r>4$ and $n>1$ be positive integers. Let $$k=⌊r^{n²}α⌋$$

where $⌊.⌋$ is the floor function.

My question is: Find the number of digits of the number $k$ in function of $r$ and $n$.

Answer 1

The number of digits of $k$ is $\lfloor \log_{10} k \rfloor +1$. For example, $\log_{10} 1234=3+$ and $1234$ has $4$ digits. So (leaving aside the floor for now) $\log_{10} k= \log_{10}(r^{n^2}\alpha)=n^2\log_{10}r+\log_{10}\alpha$ $$1+\lfloor \log_{10} k\rfloor= 1+\lfloor n^2\log_{10}r+\log_{10}\alpha\rfloor$$

Answer 2

Looks like $\lceil\log_{10}(r^{n^2}\alpha)\rceil=\lceil n^2\log_{10}r+\log_{10}\alpha\rceil$ to me.