I have an equation that looks like $x+(\ln3)y+z=0$ where there's a natural logarithm as a coefficient. Is it possible to have this in a linear equation? I know that you cannot have a root or a product of variables in a linear equation, but I'm not so sure about coefficients that include an exponent.
Yes, you can. You should distinguish parameters/coefficients and variables.
Simple test which is in fact a definition of a linear equation is the following. Let you have an equation $$ f(x,y,z) = 0 $$ e.g. in your case $f(x,y,z) = x+(\log 3)y+z$. To check it's linearity it's necessary and sufficient to have: $$ f(\alpha x'+\beta x'',\alpha y'+\beta y'',\alpha z'+\beta z'') = \alpha f(x',y',z')+\beta f(x'',y'',z''). $$