Let's simplify the given logarithmic expressions step by step using the rules of logarithms.

Part (a):

$\log_a(xy) - \log_a \sqrt[3]{x} + 2\log_a x$

Step 1: Apply the product rule to $\log_a(xy)$:

$\log_a(xy) = \log_a x + \log_a y$

So the expression becomes: $\log_a x + \log_a y - \log_a \sqrt[3]{x} + 2\log_a x$

Step 2: Apply the power rule to $-\log_a \sqrt[3]{x}$:

$\log_a \sqrt[3]{x} = \log_a x^{1/3} = \frac{1}{3}\log_a x$

Thus, the expression becomes: $\log_a x + \log_a y - \frac{1}{3}\log_a x + 2\log_a x$

Step 3: Combine like terms for $\log_a x$:

$\log_a x \left(1 - \frac{1}{3} + 2\right) + \log_a y$

Simplify the coefficient of $\log_a x$: $1 - \frac{1}{3} + 2 = \frac{3}{3} - \frac{1}{3} + \frac{6}{3} = \frac{8}{3}$

So the final simplified expression is: $\frac{8}{3}\log_a x + \log_a y$

Part (b):

$9\log_b x - 4\log_b y$

Step 1: Apply the power rule to both terms:

$9\log_b x = \log_b x^9, \quad 4\log_b y = \log_b y^4$

Substitute these into the expression: $\log_b x^9 - \log_b y^4$

Step 2: Apply the quotient rule:

$\log_b x^9 - \log_b y^4 = \log_b\left(\frac{x^9}{y^4}\right)$

So the final simplified expression is: $\log_b\left(\frac{x^9}{y^4}\right)$

Final Answers:

(a) $\frac{8}{3}\log_a x + \log_a y$
(b) $\log_b\left(\frac{x^9}{y^4}\right)$

Would you like further clarifications or detailed examples?
Here are 5 questions to expand the topic:

1. What are the key rules of logarithms, and how are they applied in simplifications?
2. How can the change of base formula help simplify logarithmic expressions?
3. Can logarithms with fractional bases (e.g., $\log_{1/2}$) be simplified similarly?
4. How are logarithmic properties used in solving exponential equations?
5. What are the applications of logarithmic simplifications in real-world problems?

Tip: Always rewrite logarithmic expressions step-by-step to avoid missing simplification opportunities!