https://file.aiquickdraw.com/mathbot/file/1727252654951ub691l9h.jpg

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.

Use the Product of Powers Property of Exponents to simplify log_b(b^x * b^y).

Learn how to simplify the logarithmic expression log_b(b^x * b^y) using the Product of Powers Property of Exponents. This step-by-step guide covers key concepts in logarithms and exponents, making it perfect for high school algebra students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation