Relation between log function and its characteristic g

AI Thread Summary
The discussion centers on the inequality g ≤ log(n) < g + 1, where g is an integer, n is a natural number, and the logarithm is base b > 1. Participants agree that this inequality indicates that log(n) falls between two consecutive integers for any positive integer n. The graphical representation of the logarithmic function supports the intuitive understanding of this relationship. The conversation highlights that this property is evident and does not necessarily require formal proof. Overall, the participants find the connection between the log function and its characteristic g to be clear and self-evident.
elliti123
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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 ?
 
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elliti123 said:
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.
 
Yes i just looked at the graph it does seem to come pretty natural to mind. Thanks for the hint.
 
?? Every number, x, satisfies n\le x&lt; n+ 1 for some integer n.
 
HAHA i can't believe i did not look at it like that. Thanks.
 

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