Answer
$$\log_b\Bigg(\frac{x^2y}{z^2}\Bigg)=2\log_bx+\log_by-2\log_bz$$
Work Step by Step
$$A=\log_b\Bigg(\frac{x^2y}{z^2}\Bigg)$$
First, we apply the Quotient Rule, which states $$\log_b\frac{M}{N}=\log_b M-\log_bN$$ ($M, N, b\in R, M\gt0, N\gt0, b\gt0, b\ne1$)
That means, $$A=\log_b(x^2y)-\log_b(z^2)$$
Now, for $\log_b(x^2y)$, we apply the Product Rule, which states $$\log_bMN=\log_bM+\log_bN$$ ($M, N, b\in R, M\gt0, N\gt0, b\gt0, b\ne1$)
Therefore, $$A=\log_b(x^2)+\log_b y-\log_b(z^2)$$
Finally, for $\log_b(x^2)$ and $\log_b(z^2)$, Power Rule can be applied, $$\log_bM^p=p\log_bM$$ ($M, b, p\in R, M\gt0, b\gt0, b\ne1$)
So, $$A=2\log_bx+\log_by-2\log_bz$$
