The function $\log \left(x-3\right)$ is 

Suppose that the function $f\left(x\right)=\log \left(x-3\right)$

         $\begin{array}{rcl}f\left(-x\right)& =& \log \left(-x-3\right)\\ & =& \log \left(x+3\right)\\ & \ne & f\left(x\right)\end{array}$

If $f\left(-x\right)=f\left(x\right)$ then the function $f\left(x\right)$ is called even function and if $f\left(-x\right)=-f\left(x\right)$ then the function $f\left(x\right)$ is called odd function 

But for the given funciton $f\left(x\right)=\log \left(x-3\right)$,

 $f\left(-x\right)$ is not equal to either $f\left(x\right)$ or $-f\left(x\right)$

Therefore, the function $f\left(x\right)=\log \left(x-3\right)$ is neither even nor odd function.