https://file.aiquickdraw.com/mathbot/file/172581718223009syfo9e.jpg

The equation visible in the image is:

\[
\log_3 x = 4 \log_3 a + 7 \log_3 b
\]

This is a logarithmic equation. To solve for \( x \), you can use logarithmic properties, specifically the following:

1. **Product rule**: \( \log_b(x) + \log_b(y) = \log_b(xy) \)
2. **Power rule**: \( n \log_b(x) = \log_b(x^n) \)

### Steps to solve the equation:

1. Apply the **Power rule** to both \( 4 \log_3 a \) and \( 7 \log_3 b \):
   \[
   \log_3 x = \log_3 a^4 + \log_3 b^7
   \]

2. Apply the **Product rule** to combine the terms on the right-hand side:
   \[
   \log_3 x = \log_3 (a^4 \cdot b^7)
   \]

3. Since the logs are the same base, you can set the arguments equal to each other:
   \[
   x = a^4 \cdot b^7
   \]

Thus, the solution for \( x \) is:
\[
x = a^4 \cdot b^7
\]

---

Would you like any further explanation on the logarithmic rules or steps? Here are some related questions to deepen your understanding:

1. How do the product and power rules in logarithms differ from each other?
2. Can you solve logarithmic equations with different bases?
3. What would happen if the logs had different coefficients or bases?
4. How can logarithmic properties be used to simplify complex logarithmic expressions?
5. How do logarithms relate to exponential functions?

**Tip**: Always check if the base of the logarithms in an equation is the same before applying rules!

Learn how to solve the logarithmic equation log_3 x = 4 log_3 a + 7 log_3 b using the power and product rules of logarithms. Step-by-step explanation suitable for students in Grades 9-12.

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