Relation between log function and its characteristic g

In summary, the conversation discusses an inequality involving logarithms and natural numbers, and the question of whether it holds true for all integer values of g. It is suggested that this can be easily observed by looking at the graph of a logarithmic function.
  • #1
elliti123
19
0
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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  • #2
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.
 
  • #3
Yes i just looked at the graph it does seem to come pretty natural to mind. Thanks for the hint.
 
  • #4
?? Every number, x, satisfies [itex]n\le x< n+ 1[/itex] for some integer n.
 
  • #5
HAHA i can't believe i did not look at it like that. Thanks.
 

1. What is the log function and its characteristic g?

The log function is a mathematical function that calculates the logarithm of a number. The characteristic g is the integer part of the logarithm, while the mantissa is the decimal part.

2. How is the log function related to its characteristic g?

The log function and its characteristic g are related through the equation log(x) = g + log(m), where x is the number, g is the characteristic, and m is the mantissa.

3. What is the significance of the characteristic g in the log function?

The characteristic g in the log function determines the placement of the decimal point in the logarithm result. It also helps to simplify complex calculations involving logarithms.

4. How does the characteristic g affect the graph of the log function?

The characteristic g shifts the graph of the log function horizontally, depending on its value. A higher value of g results in a rightward shift, while a lower value of g results in a leftward shift.

5. Can the characteristic g be negative?

Yes, the characteristic g can be negative. This occurs when the number is less than 1, and the logarithm result is a negative number.

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