Recursive Definition for Logarithm

Question

Is it possible to express $\log(n)$ in terms of itself, using only elementary functions, e.g., similar to (the incorrect made-up equation): \begin{equation*} \log(n) = e^n * \log(n - 1) + \sin(n) * \log(n - 2) \end{equation*} I know this can be done for e.g. $\sin$ and $\cos$, since \begin{equation*} \sin(n) = \sin(n - 1) * \cos(1) + \cos(n - 1) * \sin(1) \end{equation*} and \begin{equation*} \cos(n) = \cos(n - 1) * \cos(1) - \sin(n - 1) * \sin(1) \end{equation*} This question is more of theoretical interest to me.

Answer 1

$$\log ab = \log a + \log b$$

Before the advent of digital computers, that was the formula used to build up log tables very rapidly.