From the image, the question is about verifying and simplifying the logarithmic expression:

$\log_b (ab)^x = x \log_b a + x$.

Analysis:

1. Using the logarithmic power rule: $\log_b (ab)^x = x \cdot \log_b (ab)$

2. Applying the logarithmic product rule: $\log_b (ab) = \log_b a + \log_b b$

   So, the equation becomes: $x \cdot (\log_b a + \log_b b)$

3. Distributing $x$: $x \cdot \log_b a + x \cdot \log_b b$

4. The final result: $x \log_b a + x \log_b b$

Comparing this with the options in the image, the correct simplification is: $x \cdot \log_b a + x \cdot \log_b b$, which matches one of the multiple-choice answers.

Do you have any questions, or would you like a detailed explanation of each logarithmic property used?

Here are 5 related questions to expand your understanding:

1. What are the logarithmic product and power rules in general?
2. Can you explain the difference between expanding and condensing logarithms?
3. How do logarithms behave when the base is different for terms in the product?
4. Can logarithmic properties be used to solve exponential equations?
5. How can this simplification help in solving real-world logarithmic problems?

Tip: Always remember