There really isn’t all that much to this problem. We know that we can’t have negative numbers or zero in a logarithm so we’ll need to require that whatever \(\left( {x,y} \right)\) is it will need to satisfy,

\[2x - 3y + 1 > 0\]

Since this is the only condition we need to meet this is also the domain of the function.

Let’s do a little rewriting on this so we can attempt to sketch the domain.

\[2x + 1 > 3y\hspace{0.5in}\Rightarrow \hspace{0.5in}y < \frac{2}{3}x + \frac{1}{3}\]

So, it looks like we need to be below the line above. The domain is illustrated by the green area in the sketch below.

Note that we dashed the graph of the “bounding” line to illustrate that we don’t take points from the line itself.