The equation that you have derived is correct. For a zeroth-order reaction, $$\log t_{1/2}=\log a-\log2K$$
By superficially observing, this seems to be giving a straight line with the equation $y=mx-c$ but actually there are several(not all) reactions in which the $K$ value is very small. In case of any reaction having $K\lt0.5$, $2K$ becomes less than $1$ and hence $\log2K$ becomes negative. Then you would obtain an equation of the form $y=mx+c$ whose graph will have a positive ordinate and it will appear similar to the graph in the question. So that graph in your question is correct for a zero-order reaction having a small rate constant (less than $0.5$).