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Is ln(√e^π)/π a rational number?

  1. Sep 6, 2014 #1
    is ln(√eπ)/π a rational number?

    where π =3.14...
     
  2. jcsd
  3. Sep 6, 2014 #2

    WWGD

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    Wow, e alone not hard-enough? You could write a whole paper, if not a small book in answering this.
     
  4. Sep 6, 2014 #3
    Isn't it a half?
     
  5. Sep 6, 2014 #4

    WWGD

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    You may be right; I guess I jumped the gun.
     
  6. Sep 6, 2014 #5
    but my textbook says its an irrational number, how can that be?
     
  7. Sep 6, 2014 #6

    PeroK

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    Your textbook could be wrong!
     
  8. Sep 6, 2014 #7
    $$
    \begin{align}

    \frac{ \ln{ \sqrt{ e^\pi } } }{\pi} &= \frac{ \ln{ e^\frac{\pi}{2} } }{\pi}\\

    &= \frac{1}{\pi} \frac{\pi}{2}\\

    &= \frac{1}{2}

    \end{align}
    $$
    The ## \sqrt{e^\pi} ## is equivalent to ##e^{\frac{\pi}{2}}##, so the natural log cancels with ##e## and you're left with ##\frac{(\frac{\pi}{2})}{\pi}## which is ##\frac{1}{2}##.
     
    Last edited: Sep 6, 2014
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