Given that \(u = \tanh x\), use the definition of \(\tanh x\) in terms of exponentials to show that
$$x = \frac { 1 } { 2 } \ln \left( \frac { 1 + u } { 1 - u } \right)$$
Solve the equation \(4 \tanh ^ { 2 } x + \tanh x - 3 = 0\), giving the solution in the form \(a \ln b\) where \(a\) and \(b\) are rational numbers to be determined.
Explain why the equation in part (b) has only one root.