Can the expression $\log(\frac{x}{y^x}y^{(x-1)})$ be evaluated to equal $\log x - \log y^x +(x-1) \log y$ or $\log(xy^{-1})$?
I am not sure which one is correct. This question is quite trivial but I am confused which answer is correct. In my opinion the first expression is correct. (This is based on an argument with a friend of mine.)
These answers are both equivalent as $$\log{x}-x\log{y}+(x-1)\log{y}=\log{x}-\log{y}$$ $$\log{(xy^{-1})}=\log{x}+\log{y^{-1}}=\log{x}-\log{y}$$
Both are correct. Note that $x\dfrac{y^{x-1}}{y^x}=xy^{-1}$ and that therefore it is trivial that your expression is equal to $\log(xy^{-1})$. For the other expression, use that fact that $\log$ maps products and quotients into sums and subtractions respectively.
It is $$\log\frac{x}{y^x}+\log(y^{x-1})=\log(x)-x\log(y)+(x-1)\log(y)=\log(x)-\log(y)=\log\frac{x}{y}$$
