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Relation between log function and its characteristic g

  1. Jul 7, 2015 #1
    I have come across this inequality:$$ g≤ log\ n <g + 1$$
    We assume that the base of the log is b >1 and n is all the natural numbers. I would like to know if anyone could provide a proof regarding this and mention for what g ? Is it for all the g which are integers ?
  2. jcsd
  3. Jul 7, 2015 #2


    Staff: Mentor

    It seems to me that this is saying that, for any positive integer n, log(n) lies between two other integers, g and g + 1. For example, ##2 \le \log_{10}(100) < 3## and ##1 \le \log_{10}(13) < 2##. If you look at the graph of ##y = \log_b(x)##, this seems pretty obvious, and isn't something that would require a proof.
  4. Jul 7, 2015 #3
    Yes i just looked at the graph it does seem to come pretty natural to mind. Thanks for the hint.
  5. Jul 7, 2015 #4


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    Science Advisor

    ?? Every number, x, satisfies [itex]n\le x< n+ 1[/itex] for some integer n.
  6. Jul 7, 2015 #5
    HAHA i cant believe i did not look at it like that. Thanks.
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