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Calaculate a Ln without calculator

  1. Oct 6, 2012 #1
    Hello
    Im solving a problem with calculator to find the zeta in this equation

    gif.latex?e^{(\frac{-\xi\pi%20}{\sqrt{1-\xi^{2}}})}=.163.gif

    but as i see some persons they solved this equation with different numbers(1.63 ) without calculator.now i want to know
    how can fine zeta in this equation without calculator and fast?
     
    Last edited by a moderator: Apr 16, 2017
  2. jcsd
  3. Oct 6, 2012 #2

    mfb

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    2016 Award

    Staff: Mentor

    ln(.163) = -2 + ln(e^2*0.163) ≈ -2 + ln(7*0.163) ≈ -2 + ln(1.1) ≈ -2 + 0.1 = -1.9
    ##-z\pi \approx -1.9 \sqrt{1-z^2}##
    => z>0
    ##z^2 \pi^2 \approx 3.6 - 3.6 z^2##
    ##z^2 (10+3.6) \approx 3.6##
    ##z^2 \approx \frac{3.6}{13.8} \approx \frac{4}{15} \approx \frac{2.7}{10} = 0.27##
    ##z \approx 0.5##
    Solving speed ≈ typing speed
    That is very close to the exact value and indicates that the .163 and the formula have some special meaning.

    Edit: Fixed small error.
     
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