How have the moduli signs disappeared in the following step:
$$\frac1{k}\left(\ln|g+kv| - \ln|g+ku|\right) = -t$$
Therefore $$ \ln\left(\frac{g+kv}{g+ku}\right) = -kt$$
$g$, $k$ and $u$ are positive constants. $t$ is time, $v$ is velocity.
Context: the above calculations are from solving the equation $dv/dt = -g - kv$ given that $v = u$ when $t = 0$, and that $u$, $g$ and $k$ are positive constants.
If $k$, $g$, $u$ are all positive, you need to have $v$ large enough (i.e., not too negative) to make $g + k v > 0$. Maybe that results from some part that you haven't told us...