We can use the following logarithm rules:
[tex]\begin{gathered} \log _a(xy)=\log _a(x)+\log _a(y)\Rightarrow\text{ Product rule} \\ \log _a(\frac{x}{y})=_{}\log _a(x)-\log _a(y)\Rightarrow\text{ Quotient rule} \end{gathered}[/tex]
Then, we have:
[tex]\begin{gathered} \text{ Apply the product rule} \\ \log (\frac{x}{z}w)=\log (\frac{x}{z})+\log (w) \\ \text{ Apply the quotient rule} \\ \log (\frac{x}{z}w)=\log (x)-\log (z)+\log (w) \end{gathered}[/tex]
Therefore, the given expression expanded as a sum or difference of logs is:
[tex]\log (x)-\log (z)+\log (w)[/tex]
