Q.Express $\cosh^{-1}\left(\dfrac{5}{4}\right)$ in logarithmic form and hence find its value.

Every inverse hyperbolic function can be rewritten as an explicit natural logarithm, and this is what makes the inverse functions computable rather than merely abstract symbols. The three principal theorems are:

The proof strategy is identical in every case: set $y$ equal to the inverse function, write $x$ in terms of $e^y$ and $e^{-y}$ using the original definition, and multiply through by $e^y$ (or $e^{2y}$) to turn the equation into a genuine quadratic in $e^y$. Solving that quadratic, exactly one root is admissible (the other is rejected because $e^y$ must be positive, or because it lies outside the principal branch), and taking a logarithm of the surviving root recovers $y$.

For instance, for $\sinh^{-1}x$, writing $x=\sinh y=\tfrac12(e^y-e^{-y})$ gives $e^{2y}-2xe^y-1=0$; the quadratic formula gives $e^y=x\pm\sqrt{x^2+1}$, and since $\sqrt{x^2+1}>|x|$ always, only $e^y=x+\sqrt{x^2+1}$ is positive for every real $x$ — so no domain restriction is needed here, unlike for $\cosh^{-1}x$, where both roots of the corresponding quadratic are positive when $x>1$ and we must additionally insist on the branch $y\ge0$ to pick $x+\sqrt{x^2-1}$ over its reciprocal $x-\sqrt{x^2-1}$.

These log-forms turn every "evaluate $\sinh^{-1}(\ldots)$" or "solve $\cosh x = \ldots$" problem into ordinary algebra: substitute the given number into the formula, simplify the surd, and (if a decimal is wanted) take the logarithm at the end — never before combining the terms inside it.