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It's famously unknown whether the natural log of 2 is rational or not.

How about the natural log of other numbers. Is it known/unknown whether these are rational?

Obviously ln(1) is 0, and ln(2^n) is n*ln(2) (and is thus rational iff ln(2) is rational), but how about other cases?

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Yes, Virginia, this is related. –  J. M. Dec 23 '10 at 3:38
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$\log(2)=\frac{p}{q}\Rightarrow e^p=2^q$. Because $e$ is not algebraic, $\log(2)$ is not rational. (Willie Wong's answer includes this and much more, but I thought a comment up here countering the first sentence might help.) –  Jonas Meyer Dec 23 '10 at 3:40
    
Thanks Jonas, I was a little confused by that statement. –  picakhu Dec 23 '10 at 3:43
    
For some reason, I thought no one knew whether the sum of the alternating harmonic sequence (ln(2)) was rational or not. Guess I was wrong! –  barrycarter Dec 23 '10 at 4:39
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@barrycarter: Perhaps you were thinking of en.wikipedia.org/wiki/Euler%E2%80%93Mascheroni_constant. @picakhu: Proving that $e$ is irrational isn't too hard. Concerning the difficulty of the proof that it is transcendental, you may be interested in the following if you haven't seen it: math.stackexchange.com/questions/12872/… –  Jonas Meyer Dec 23 '10 at 5:09
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2 Answers 2

up vote 8 down vote accepted

To shoot a bird with a cannon...

By the Lindemann-Weierstrass theorem, $e^a$ is transcendental for all $a$ algebraic and non-zero. In particular if $a$ is rational, $e^a$ cannot be rational. Hence $\ln(n)$ is always irrational.

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(Yes, that answer is given somewhat tongue in cheek. If there exists a $r\in\mathbb{Q}$ such that $e^r = n$, then raising to a higher power we have that there exists $p,q\in \mathbb{N}$ such that $e^p = q$. This will imply that $e$ is algebraic. But we know $e$ is transcendental.) –  Willie Wong Dec 23 '10 at 3:32
    
why is that shooting a bird with a canon? Answering a whole class of problems in one go and not to bother to deal with each individual case should be applauded. So what if some people's elephant is reduced to a fly? –  Arjang Dec 23 '10 at 3:33
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@Arjang: see my comment above. It suffices in reality to know that $e$ is transcendental, a result which predates the Lindemann-Weierstrass theorem. I just like to mention the LW theorem when there is a chance (as if that is not common enough on math websites). –  Willie Wong Dec 23 '10 at 3:37
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Sometimes, it's fun to nuke mosquitoes... :D –  J. M. Dec 23 '10 at 3:57
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We can also use a non-simple continued fraction expansion of $\displaystyle e^{2x/y}$ to prove the irrationality of $\displaystyle e^{2x/y}$ when $\displaystyle x,y$ are positive integers. Thus if $\displaystyle \log n = x/y$, then $\displaystyle e^{2x/y} = n^2 $ is rational, contradicting irrationality of $\displaystyle e^{2x/y}$.

Incidentally, the first proof of irrationality of $\pi$ by Lambert used a continued fraction expansion (of $\tan x$, I believe).

The expansion we use:

alt text

and the theorem we use to prove irrationality is quoted in the wiki page for Generalized Continued Fractions here: Conditions of Irrationality.

By this theorem, it is enough that for all sufficiently large positive integers $\displaystyle m$ we have that $\displaystyle (2m+1)y \gt x^2$, which is true for fixed $\displaystyle x,y$.

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