Approximations and limits for Lambert $W$ of decreasing functions

Question

I'm aware of the following approximation that approaches the Lambert W Function: $$ W(f(x))\approx\ln\left(f(x)\right)-\ln\left(\ln(f(x))\right) $$ However, this approximation fails to capture the Lambert $W$ function if $f(x)$ is a decreasing function. Is there an approximation, potentially in terms of logs (though not necessarily) that approaches the same limit as the product log of a decreasing function? Thanks.