$y = \ln(e^x)$

$y = \ln(e^x)$

$y = \ln(e^x - 1)$

$y = \ln(e^x)$ is incorrect. $y = \ln(e^x)$ actually simplifies to $x$, but $y(0)=0$. Correct: $y = \ln(e^x)$? No, $y = \ln(e^x)$ yields $y' = 1$, whereas $e^{x-y} = e^{x-x} = 1$. Both match. Actually $y = \ln(e^x) = x$.