Problem

(a) Using the appropriate definitions of hyperbolic functions in terms of exponentials, show that

$$\sinh x + \sech x = \frac{e^{4x} + 4e^{2x} - 1}{2e^{3x}}$$

(b) Hence solve the equation

$$\frac{1}{2}\sinh x + \sech x = e^x$$

Give the answer in the form $a \ln b$ where $a$ is a rational number and $b$ is an integer.

(Solutions relying on calculator technology are not acceptable.)