Logarithmically enhanced nonlinear boundary layer ($x=O(\epsilon \log (1/\epsilon ))$)

If a leading outer expansion diverges as $\log x$ and a derivative coefficient contains $x-\epsilon z$, balancing that nonlinear correction gives a layer larger than $O(\epsilon )$. Write $x=\epsilon LX$ with $L=\log (1/\epsilon )$ and subtract the large negative constant from $z$. Matching often introduces a $\log L$ correction and yields an endpoint slope of order $1/(\epsilon L)$. The balance must be derived from the actual equation; logarithmic divergence alone does not fix a universal layer width.