The hyperbolic sine function, \sinh x, is one-to-one, and therefore has a well-defined inverse, \sinh^{-1} x, shown in blue in the figure. In order to invert the hyperbolic cosine function, however, we need (as with square root) to restrict its domain. By convention, \cosh^{-1} x is taken to mean the positive number y such that x=\cosh y. This function is shown in red in the figure; notice that \cosh^{-1} x is defined only for x\ge 1 (at least where real numbers are concerned).
Figure 1: Plots of \cosh^{-1} x (red), \sinh^{-1} x (blue) and \tanh^{-1}
x (green)
The hyperbolic tangent function is also one-to-one and invertible; its inverse, \tanh^{-1} x, is shown in green. It is defined only for -1< x<1.
Just as the hyperbolic functions themselves may be expressed in terms of exponential functions, so their inverses may be expressed in terms of logarithms. Taking the case of \sinh first, suppose
\begin{eqnarray} x-\sqrt{x^2-1}&=&(x-\sqrt{x^2-1})\times\frac{x+\sqrt{x^2-1}}{x+\sqrt{x^2-1}}\\ &=&\frac{x^2-(x^2-1)}{x+\sqrt{x^2-1}}\\ &=&\frac{1}{x+\sqrt{x^2-1}}, \end{eqnarray}
and therefore
\begin{eqnarray} \ln(x-\sqrt{x^2-1})&=&\ln\left(\frac{1}{x+\sqrt{x^2-1}}\right)\\ &=&-\ln(x+\sqrt{x^2-1}). \end{eqnarray}
So our solutions are