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The question in the image asks to evaluate the following expression and determine for which values of \( x \) the expression is valid:

\[
8 \log_6(6x) - \log_6 \left( x^8 \right) = 8
\]

### Step 1: Simplify the expression
Start by simplifying the logarithmic terms:

1. **For the first term**, apply the logarithmic property \(\log_b(ab) = \log_b a + \log_b b\):
   \[
   \log_6(6x) = \log_6(6) + \log_6(x)
   \]
   Since \(\log_6(6) = 1\), the first term simplifies to:
   \[
   8 \log_6(6x) = 8(1 + \log_6(x)) = 8 + 8\log_6(x)
   \]

2. **For the second term**, use the logarithmic property \(\log_b(x^n) = n\log_b(x)\):
   \[
   \log_6(x^8) = 8\log_6(x)
   \]

So the entire expression becomes:
\[
8 + 8\log_6(x) - 8\log_6(x) = 8
\]

### Step 2: Analyze for which values of \( x \) the expression is valid
Both logarithmic expressions, \(\log_6(6x)\) and \(\log_6(x^8)\), require that \(x > 0\), since the logarithm of a non-positive number is undefined.

### Conclusion:
- The expression is valid for \( x > 0 \) because logarithms are only defined for positive values of \( x \).
- Therefore, **option B** ("The value of this expression is valid for \( x > 0 \), since \( \log_6(x^8) \) is not defined for \( x \leq 0 \)") is the correct answer.

Would you like further clarification or a more detailed explanation?

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Here are five questions based on this topic:

1. What are the main logarithmic properties used to simplify expressions?
2. Why is the logarithmic function undefined for non-positive values?
3. How can you use logarithmic properties to solve exponential equations?
4. Can the base of a logarithm affect the domain of its argument?
5. What are the conditions for solving logarithmic equations involving products or powers?

**Tip:** Always check the domain of a logarithmic function to avoid undefined values when simplifying or solving equations!

Evaluate 8 log_6(6x) - log_6(x^8). For what values of x is the value of this expression valid?

Learn how to evaluate and simplify the logarithmic expression 8 log_6(6x) - log_6(x^8) and determine for which values of x it is valid. This problem involves applying logarithmic properties and is suitable for students in Grades 10-12.

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