The problem in the image asks to simplify using the Product of Powers Property of Exponents.

Here’s the expression:

$\log_b(b^x \cdot b^y)$

Step-by-step simplification:

1. Apply the Product of Powers Property: The property states that when multiplying two powers with the same base, you add their exponents. Therefore:

   $b^x \cdot b^y = b^{x + y}$

2. Substitute this into the logarithm:

   $\log_b(b^{x + y})$

3. Simplify the logarithmic expression: By the basic logarithmic identity, $\log_b(b^a) = a$, where $a$ is the exponent:

   $\log_b(b^{x + y}) = x + y$

Final Answer:

$\log_b(b^x \cdot b^y) = x + y$

This is the simplified form of the expression.

Would you like further details or have any questions?

Here are 5 related questions to explore this concept further:

1. How would you simplify $\log_b(b^2 \cdot b^5)$?
2. What is the logarithmic identity used in this simplification?
3. How does the power rule of logarithms differ from the product rule?
4. What is the result of $\log_b(b^{x-y})$?
5. Can this property be extended to more than two terms, like $\log_b(b^x \cdot b^y \cdot b^z)$?

Tip:

When working with logarithmic expressions, always look for opportunities to apply logarithmic identities like $\log_b(b^a) = a$ to simplify your calculations.