Relation between log function and its characteristic g

[elliti123]

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 ?

 

[Mark44]

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 ?

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.

 

[elliti123]

Yes i just looked at the graph it does seem to come pretty natural to mind. Thanks for the hint.

 

[HallsofIvy]

?? Every number, x, satisfies $n\le x< n+ 1$ for some integer n.

 

[elliti123]

HAHA i can't believe i did not look at it like that. Thanks.