help me find the natural logarithum of csch^-1 ((-1)/(2 sqrt(30)))

Question

arindameducationusc · Answer

Hyperbolic cosecant of x = csch x = 2/(e^x - e^-x)

irishboy123 · Answer

$\large ln [csch^{-1} (\frac{-1}{2 \sqrt(30)})]$ ??

i think there may be a misunderstanding in the question or on my part which this might clarify

$\large cosech(y) =  (\frac{-1}{2 \sqrt(30)})  = \frac{2}{e^y - e^{-y} }$

let $ (\frac{-1}{2 \sqrt(30)}) = \alpha $

$\large \alpha (e^y - e^{-y}) = 2$

$\large \alpha e^{2y} - 2e^y - \alpha = 0$

let  $\large z =e^y $

$\large \alpha z^2 - 2 z - \alpha = 0$

$\large z = \frac{2 \pm \sqrt{4 + 4 \alpha^2}}{2 \alpha} = \frac{1 \pm \sqrt{1 +  \alpha^2}}{ \alpha}$

$\large y = ln [\frac{1 \pm \sqrt{1 +  \alpha^2}}{ \alpha}] = ln [\frac{1}{\alpha } + \sqrt{1 +  \frac{1}{\alpha^2}}]$

the last tweak should ensure we get the positive number for the log

and so the question is this: isn't the log already in the inverse cosech or are you taking a further log?