Why is the labour supply independent of the wage in this case? Consider the model $\max U = \ln c + \ln l$ subject to $c=Lw$ and $1=l+L$, where $c$ denotes consumption, $l$ leisure, $L$ the labour and $w$ the wage rate.

The optimal choices of $L$ and $l$ can be easily computed to be 1/2. I don't understand the intuition why the labour supply decision is independent of the wage rate in this case.
Is it possible to argue in terms of substitution effect and income effect of a wage rate change?
For some intuition, rewrite the problem as the consumption problem $\max_{(c, l)\in\mathbb{R}_{+}^{2}} (cl)^{0.5}$ subject to the constraint $c/w + l \leq 1$.
The optimal level of $l$ is then given by $l = 1/2$ as per the usual solution for Cobb-Douglas preferences.
The intuition for why $l$ does not depend on $w$ in this example is the same intuition for Cobb-Douglas consumers, who spend a fixed fraction of their income on each good.