I'm working through an introduction to differential equations and we saw this very basic equation (let's say $x$ is in dollars and $t$ is in years:
$\dot{x}(t) - kx(t)=-a$
Without running through all the details, I solved this and got
$x(t)=\frac{a}{k}+(x_0-\frac{a}{k})e^{kt}$
Okay, so far, I understand the units.
(1) $a$ is in dollars per year and $k$ is in 1/year so $a/k$ is in units of dollars.
(2) $x_0$ is in units of dollars so $(x_0-\frac{a}{k})$ is in units of dollars.
(3) $k$ is in units of 1/year and $t$ is in units of year so the exponential has no units.
My problem comes when I try to solve for $x(t)=0$
$0=\frac{a}{k}+(x_0-\frac{a}{k})e^{kt}$
$(\frac{a}{k}-x_0)e^{kt}=\frac{a}{k}$
$e^{kt}=\frac{a}{a-kx_0}$
$kt=ln(\frac{a}{a-kx_0})$
$t=\frac{1}{k}ln(\frac{a}{a-kx_0})$
Here, I'm still good - inside the log the units cancel out and $k$ is 1/years so $1/k$ is in units of years.
But, now for fun, I use the $ln$ rules and get:
$t=\frac{1}{k}[ln(a)-ln(a-kx_0)]$
And this is where I get confused. Inside each natural log, I get units of dollars/year - but how do you interpret the units?