Explanation

(c)
$x\log_{e}{a + \frac{x^{3}}{3!}\left( \log_{e}a \right)^{3} + \frac{x^{5}}{5!}\left( \log_{e}a \right)^{5} + ...}$
$= \frac{e^{x}\log_{e}{a - e^{- x}\log_{e}a}}{2}\ \ \ \ \ \ \ \left\lbrack \because\ \frac{e^{x} - e^{- x}}{2} = x + \frac{x^{3}}{3!} + \frac{x^{5}}{5!} + ... \right\rbrack$
$= \sinh{(x\log_{e}a)}$