Get arsinh from sinh

Question

I need to establish the inverse function of the hyperbolic sine: I am trying to do this by setting $y = \sinh(x)$ and solving for $x$, however I got stuck at this: $$ y=\frac{e^x -e^{-x}}{2} $$ $$ 2y=e^x - e^{-x} $$ I dont know how to solve for x at this point, though. Taking the logarithm seems nonsensical with a sum on the right side. The end goal is the arsinh given by $$ y = \log(x + \sqrt{x^2 +1}) $$

Answer 1

Hint: make replacement $e^x = z$ and use $e^{-x} = 1 / z$ to get quadratic equation on $z$.

Answer 2

Hint: Write $$2y=e^x-\frac{1}{e^x}$$, now substitute $$t=e^x$$ and solve the quadratic.