I am to rewrite $\ln(z)-\ln(x)-\ln(y)$ in compact form. The solution in my textbook says it's $\ln(\frac{z}{xy})$.
I'm confused by this since thought that a product, as seen in the solutions denominator, only happens when there's a summation of logs?
My attempted answer:
$$\ln\left(\frac{z}{\frac{x}{y}}\right).$$
Why is my answer incorrect and why does the solution contain a product of logs when there's only subtraction in the original question?
Best Answer
Maybe it helps seeing that: $$\ln(z)-\ln(x)-\ln(y) = \ln(z) - \Big(\ln(x)+\ln(y)\Big).$$
At the same time, I think you're thinking of it as: $$ \Big(\ln(z)-\ln(x)\Big) - \ln(y). $$ Even in this case, the term inside the $\ln(\cdot)$ function amounts to: $$ \frac{z}{x}\cdot\frac{1}{y} = \frac{z}{xy}. $$