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## Main Question or Discussion Point

Hi Everyone,

I am reading up on information theory, and every resource I have found on the topic which derives the form of entropy uses the following inequality as part of the proof.

Let [tex]n[/tex] be a fixed positive integer greater than 1. If [tex]r[/tex] is an arbitrary positive integer, then the number [tex]2^r[/tex] lies somewhere between two powers of [tex]n[/tex]. i.e. There exists a positive integer [tex]k[/tex] such that

[tex]n^k\leq 2^r < n^{(k+1)}[/tex]

However, none of them prove it and I am unclear how to do so. If anyone has any ideas on how to prove it, or topics I should look into which would make this inequality obvious I would really appreciate it.

I am reading up on information theory, and every resource I have found on the topic which derives the form of entropy uses the following inequality as part of the proof.

Let [tex]n[/tex] be a fixed positive integer greater than 1. If [tex]r[/tex] is an arbitrary positive integer, then the number [tex]2^r[/tex] lies somewhere between two powers of [tex]n[/tex]. i.e. There exists a positive integer [tex]k[/tex] such that

[tex]n^k\leq 2^r < n^{(k+1)}[/tex]

However, none of them prove it and I am unclear how to do so. If anyone has any ideas on how to prove it, or topics I should look into which would make this inequality obvious I would really appreciate it.