We know the nth derivative of $ f(x)=\log(x)$ being a clean
$$f^{(n)}(x)= - \frac{(-1)^{n}x^{-n}}{(n-1)!},$$
but when you consider that you need the chain rule, what is the $n$th derivative for $ \log(g(x))$ and without a completely impractical$\binom{n}{k}$thrown in, only just algebraic operations and factorials of a single parameter just like the original $n$th derivative of $ \log(x)$?