https://file.aiquickdraw.com/mathbot/file/1729480771018lsvcyre0.jpg

To solve this logarithmic expression \(\log_b (y^5x)\), we will apply the properties of logarithms.

### Steps:
1. **Use the product rule** of logarithms, which states: 
   \[
   \log_b (AB) = \log_b A + \log_b B
   \]
   Applying this to \(\log_b (y^5x)\):
   \[
   \log_b (y^5x) = \log_b (y^5) + \log_b x
   \]

2. **Use the power rule** of logarithms, which states:
   \[
   \log_b (A^n) = n \log_b A
   \]
   Applying this to \(\log_b (y^5)\):
   \[
   \log_b (y^5) = 5 \log_b y
   \]

3. Putting everything together:
   \[
   \log_b (y^5x) = 5 \log_b y + \log_b x
   \]

So, the expanded form is:
\[
\log_b (y^5x) = 5 \log_b y + \log_b x
\]

Let me know if you'd like further details or have any questions!

### Related Questions:
1. How do the product and quotient rules for logarithms differ?
2. How can the change of base formula be applied to logarithms?
3. What is the inverse function of logarithms?
4. Can you simplify \(\log_b (x^2 / y^3)\)?
5. How do logarithmic properties apply to solving equations?

### Tip:
Always remember the key logarithmic rules (product, quotient, and power) as they are useful in simplifying complex expressions!

Use properties of logarithms to expand each logarithmic expression as much as possible. Evaluate logarithmic expressions without using a calculator if possible. \(\log_b (y^5x)\)

Learn how to use the properties of logarithms, such as the product and power rules, to expand and simplify the expression \(\log_b (y^5x)\). This detailed solution provides a step-by-step approach, suitable for students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation