Question

Check whether or not * is a binary operation on S={x∈R| x>0}, where x*y=| In(xy)|

Answer 1

By definition of a binary operation, we should have for any $x,y\in S$, $x*y\in S$. But if we take $x\in S, y=\frac{1}{x}$ (that always exists, as $x\neq 0$), we obtain $x*y=|\ln(x\cdot \frac{1}{x})|=|0|\notin S$. Thus the image of $*$ is not contained in $S$, so $*$ is not a binary operation.