## Calculus: Early Transcendentals 8th Edition

$z = -ln(e^t + C)$
$dz/dt + e^{t+z} = 0$ This can be rewritten as: $dz/dt = -e^{t+z}$ $dz/dt = -e^te^z$ Now let's separate z and t: $e^{-z} dz = -e^t dt$ Now let's integrate both sides: $\int e^{-z} dz = \int -e^t dt$ $-e^{-z} = -e^t + C$ To isolate z, we can add ln() on both sides: $ln(-e^{-z}) = ln(-e^t + C)$ $-z = ln(-e^t + C)$ $z = -ln(-e^t + C)$