Q.Express $\tanh^{-1}\left(\dfrac{1}{2}\right)$ in logarithmic form and hence evaluate it.

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 $y$. …