There are well known bounds for $W_0$, the "principal" real-valued branch of the Lambert-W function. For example, $W_0(x)$ lies between $\log x - \log\log x$ and $\log x - \frac{1}{2}\log \log x$, and associated big-O asymptotics of $W_0$ as $x\rightarrow\infty$.

What about bounds for $W_{-1}(x)$, the other real-valued branch of the Lambert-W function as $x\rightarrow 0^-$? The limit is well-known to be $-\infty$, but how fast does $W_{-1}(x)$ approach the limit? Are $O$ asymptotics known in terms of more elementary functions such as powers and logs?