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$M^k \le 2^r < M^{k+1}$

where $M>1 , k>0$ for some $r$.

It simply tells you that there exists a $2^r$ between $M^k$ and $M^{k+1}$. for example:

if $M=3$, $k=1$ then $$M^k = 3, \quad M^{k+1} = 9$$ and there exists $4$ and $8$ in between $3$ and $9$. i.e., $2^2$ and $2^3$


Edit: (T.B.)

Let $M \geq 2$ and $k \geq 1$ be integers. How can I prove that there exists an integer $r$ such that $$M^{k} \leq 2^r \lt M^{k+1}\quad?$$

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    This is uninterpretable, I’m afraid. What is $r$? Is $K$ the same as $k$?2011-08-13
  • 0
    In addition, this was also posted over at Overflow.2011-08-13
  • 0
    @Theo: Done. I think that your interpretation is pretty clearly right, and it makes a very reasonable question.2011-08-13
  • 0
    @Theo Done! Thanks for that -2011-08-13

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