Does the logarithm satisfy any differential equation?

If $y(x)= \log x$, then $y'(x)=\frac{1}{x}=e^{-y(x)}.$

$\ln(x)$ (and every other logarithm) satisfies the linear, homogeneous differential equation $$ xf''(x) + f'(x) = 0. $$ For $\ln$, this is admittedly just another way of stating $f'(x) = x^{-1}$.