If $f\left(x\right)$ is a function such that  $f\left(xy\right)=f\left(x\right)+f\left(y\right)$  and  $f\left(2\right)=1$ then $f\left(x\right)=$

Given that $f\left(xy\right)=f\left(x\right)+f\left(y\right)$

It implies that $f\left(x\right)={\log }_{k}x$

Given that $f\left(2\right)=1$

hence, ${\log }_{k}2=1\Rightarrow k=2$

Therefore, the required function is $f\left(x\right)=\boxed{\mathbf{l}\mathbf{o}{\mathbf{g}}_{\mathbf{2}}\mathbf{x}}$