## anonymous one year ago help me find the natural logarithum of csch^-1 ((-1)/(2 sqrt(30)))

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1. arindameducationusc

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

2. IrishBoy123

$$\large ln [csch^{-1} (\frac{-1}{2 \sqrt(30)})]$$ ?? 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?