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?

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!