Can you prove or disprove this floor/ceiling equation for positive integers \(\displaystyle k\) and \(\displaystyle b\)?
\(\displaystyle \lfloor \log_{b}{k} \rfloor + 1 = \lceil \log_{b}{\left( k+1 \right)} \rceil\)
You'll probably recognize \(\displaystyle \lfloor \log_{b}{k} \rfloor + 1\) as the number of positional notation digits of positive integer \(\displaystyle k\) in base \(\displaystyle b\).
\(\displaystyle \lfloor \log_{b}{k} \rfloor + 1 = \lceil \log_{b}{\left( k+1 \right)} \rceil\)
You'll probably recognize \(\displaystyle \lfloor \log_{b}{k} \rfloor + 1\) as the number of positional notation digits of positive integer \(\displaystyle k\) in base \(\displaystyle b\).