Is there a function such that the $n^{th}$ derivative equals the $n^{th}$ power of the first derivative

Question

I have seen functions whose second derivative is equal to the square of the first derivative. So is there a function whose $n^{th}$ derivative is equal to the $n^{th}$ power of the first derivative. Or, is there a function such that –

$$\left(\frac{df}{dx}\right)^n=\frac{d^{n}f}{dx^{n}}$$

Answer 1

$f(x) = -((n-1)!)^{1/(n-1)} \ln(x)$ is a solution.