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 Post subject: Exponential functionPosted: Sun, 27 May 2012 02:48:38 UTC
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Let be a positive irrational number and be a positive rational number. Does hold only if ? Are there any other values of that make rational?

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 Post subject: Re: Exponential functionPosted: Sun, 27 May 2012 04:19:17 UTC
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glebovg wrote:
Let be a positive irrational number and be a positive rational number. Does hold only if ? Are there any other values of that make rational?

That would be the definition of an inverse function.

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 Post subject: Re: Exponential functionPosted: Sun, 27 May 2012 04:25:58 UTC
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Are there any cases when is rational? The Lindemann-Weierstrass theorem states that if is algebraic, then is transcendental, but when is rational?

Last edited by glebovg on Sun, 27 May 2012 04:27:32 UTC, edited 2 times in total.

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 Post subject: Re: Exponential functionPosted: Sun, 27 May 2012 04:26:54 UTC
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glebovg wrote:
Are there any cases when is rational? The Lindemann-Weierstrass theorem states that if is algebraic, then is transcendental, but when is rational?

?

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 Post subject: Re: Exponential functionPosted: Sun, 27 May 2012 04:28:38 UTC
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But 0 is rational. Look at my original post. I defined to be a positive irrational number.

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 Post subject: Re: Exponential functionPosted: Sun, 27 May 2012 04:31:38 UTC
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glebovg wrote:
But 0 is rational. Look at my original post. I defined to be a positive irrational number.

No he meant that you are wrong, 0 is algebraic and is NOT transcendental, if you want to keep your original context you should say so.

In any case again, all you need is the definition of an inverse function.

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 Post subject: Re: Exponential functionPosted: Sun, 27 May 2012 04:32:29 UTC
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Hence the use of different letters. is supposed to be a nonzero algebraic number. Sorry.

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 Post subject: Re: Exponential functionPosted: Sun, 27 May 2012 04:35:36 UTC
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So, there are no other rational numbers of the form ? Is there a theorem?

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 Post subject: Re: Exponential functionPosted: Sun, 27 May 2012 04:51:04 UTC
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glebovg wrote:
So, there are no other rational numbers of the form ? Is there a theorem?

I would not call that result "theorem", it's just immediate from the definition of an inverse function, or of a 1-1 function, or of a lot of other things.

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 Post subject: Re: Exponential functionPosted: Sun, 27 May 2012 05:30:16 UTC
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So, is irrational?

Last edited by glebovg on Sun, 27 May 2012 05:52:10 UTC, edited 1 time in total.

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 Post subject: Re: Exponential functionPosted: Sun, 27 May 2012 05:31:08 UTC
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glebovg wrote:
So, is is irrational?

unless e is the log of a rational number.

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 Post subject: Re: Exponential functionPosted: Sun, 27 May 2012 06:00:46 UTC
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What do you mean? Does this also work for all ?

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 Post subject: Re: Exponential functionPosted: Sun, 27 May 2012 06:03:45 UTC
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glebovg wrote:
What do you mean? Does this also work for all ?

Assuming you're dealing with positive, real n not equal to 1.

I mean, this is really basic stuff from algebra II in high school, look at the definition of the log function. If you still cannot convince yourself use the derivative to show the exponential (respectively log) are 1-1 on their domains, there are tons of ways to see this.

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 Post subject: Re: Exponential functionPosted: Sun, 27 May 2012 07:00:05 UTC
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But why is it unknown whether is irrational?

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 Post subject: Re: Exponential functionPosted: Sun, 27 May 2012 07:01:12 UTC
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glebovg wrote:
But why is it unknown whether is irrational?

Because that's equivalent to knowing if is rational...a very difficult thing to decide in general. Understanding which numbers we can easily express are or are not algebraic is a VERY difficult prospect in general.

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