Which of the following rules are operations on the indicated set?
$$ a*b=a \ln (b) $$ on the set $$ {\{x \in \mathbb{R}: x>0}\} $$
I said no, because we can rewrite $a \ln (b) $ as $ \ln (b^a)$ and if b=1, well anything to the power of base 1 is 1 and we know $\ln(1)=0$ and x>0, therefore this cannot be an operation on the $\mathbb{R}$
Anyone can confirm my answer?
You can answer this more easily by using an example such as $a=1,\ b=e^{-1}$. In this case,
$$a*b=1 \ln (e^{-1})=-1$$
and the result is not in your set ${\{x \in \mathbb{R}: x>0}\}$. Thus that "operation" is not closed.
There are many other examples, of course, such as $a=b=1$, which I think is what you were getting at.