Where a Log-Barrier Loss Settles

You are training a model with a single positive weight $w > 0$. The loss combines a squared penalty with a log-barrier that blows up as $w$ approaches $0$:

$L(w) = w^2 - 4\ln w, \qquad w > 0$

Gradient descent comes to rest exactly where the gradient is zero.

Where does training settle, and what loss value does it reach there?