Step-by-step Solution:

Given that: \[ f(x) = \log \left(x + \sqrt{x^2 + 1} \right), \] \[ f(-x) = \log \left(-x + \sqrt{(-x)^2 + 1} \right). \] Adding both functions, we have: \[ \begin{aligned} f(x) + f(-x) & = \log \left[\left(x + \sqrt{1 + x^2}\right)\left(-x + \sqrt{1 + x^2}\right)\right] \\ & = \log \left(-x^2 + (1 + x^2)\right) \\ & = \log 1 = 0. \end{aligned} \] Thus, \[ f(-x) = -f(x), \] and hence, \( f(x) \) is an odd function.