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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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    @Theo Done! Thanks for that -2011-08-13

3 Answers 3

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The question is a variation on the well known fact that any real interval of length greater than $1$ contains at least one integer.

Hint: Assume that $a$ and $b$ are positive real numbers such that $b\ge 2a$. Then $a\le 2^r, where $r=\max R$ with $ R=\{n\ \text{integer}\mid 2^n Of course, several things need to be checked before one can call this a proof, to begin with, that $a=M^k$ and $b=M^{k+1}$ is an admissible choice, that the set $R$ is non empty and indeed has a maximum...

After all this is done, one could check that r'=\min R' works as well, with R'=\{n\ \text{integer}\mid 2^n\ge a\}.

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    @Pierre-Yves, you know, I (reluctantly) admit you (might) have (kind of) a point here... :-) Thanks! Your version is definitely better, I revised my answer.2011-08-13
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The simplest way to state this is using the natural logarithm : the inequality $ M^k \le 2^r is equivalent to $ k \log M \le r \log 2 < (k+1) \log M $ and again equivalent to $ k \left( \frac{\log M}{\log 2} \right) \le r < (k+1) \left( \frac{ \log M}{ \log 2} \right). $ which clearly has a solution because this is asking if there is an integer in the interval $[ k(\log M / \log 2), (k+1) ( \log M / \log 2 ) )$. It is possible because $M \ge 2$, hence the fraction of logarithms is greater than $1$ and the interval has length bigger than $1$.

Hope that helps,

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    Alternatively one could use the logarithm in base $2$ and see that nothing different happens.2011-08-13
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Here’s another approach. Let $\lg x$ be the log base $2$ of $x$. If $M\ge 2$, then $\lg M \ge 1$, so $\lg M^{k+1} - \lg M^k = (k+1)\lg M - k\lg M = \lg M \ge 1$. That is, the length of the interval $\left[\lg M^k, \lg M^{k+1}\right)$ is at least $1$, so the interval must contain some integer $r$. Then $\lg M^k \le r < \lg M^{k+1}$, so ...?

Added: Though they look very different, this solution and Didier’s are actually very closely related. You might find it useful to try to see why his r' = \lceil \lg M^k \rceil, and his $r = \lfloor \lg M^{k+1} \rfloor$ unless $\lg M^{k+1}$ is an integer, in which case his $r = \lfloor \lg M^{k+1} \rfloor - 1$.