Question 50
a) $\{x \in \mathbb{R} \mid \log_2[x] = [\log_2 x]\} = \bigcup_{m \in \mathbb{N}} [2^m, 2^m + 1)$.
b) $\{x \in \mathbb{R} \mid 2^{\lfloor x \rfloor} = \lfloor 2^x \rfloor\} = \bigcup_{m \in \mathbb{N}} [m, \log_2 (2^m + 1))$.
(here, $[a]$ denotes the integer part (floor function) of the real number $a$).
b) $\{x \in \mathbb{R} \mid 2^{\lfloor x \rfloor} = \lfloor 2^x \rfloor\} = \bigcup_{m \in \mathbb{N}} [m, \log_2 (2^m + 1))$.
(here, $[a]$ denotes the integer part (floor function) of the real number $a$).