log sequence limit

Consider the sequence defined as follows, with $x_1 \gt 0$.

$$x_{n+1} = \log (1 + x_n)$$

One can show that $x_n \to 0$ as $n \to \infty$ (try it).

What can you say about the sequence $y_n = nx_n$ as $n \to \infty$? Does the limit exist? If so, what is it?

Note: The $\log$ is the natural logarithm to base $e$.

[Solution]