Hello there. To solve this question, we have to remember some properties about logarithms.
Given the expression:
[tex]\ln(x)+2\ln(y)-\ln(z)[/tex]
We want to rewrite it as a single logarithm.
For this, remember the following rules:
[tex]\begin{gathered} \log_a(b)+\log_a(c)=\log_a(b\cdot c) \\ \\ \log_a(b)-\log_a(c)=\log_a\left(\dfrac{b}{c}\right) \\ \\ c\cdot\log_a(b)=\log_a(b^c) \end{gathered}[/tex]
In this case we apply the third rule to the middle logarithm in order to get:
[tex]\ln(x)+\ln(y^2)-\ln(z)[/tex]
Apply the first and second rules
[tex]\begin{gathered} \ln(xy^2)-\ln(z)\text{ \lparen First rule\rparen} \\ \\ \Rightarrow\ln\left(\dfrac{xy^2}{z}\right)\text{ \lparen Second rule\rparen} \end{gathered}[/tex]
This is the answer to this question and it is contained in the first option.
