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To find the lowest value of \( x \) that satisfies the equation, let's simplify and solve it step-by-step.

The given equation is:
\[
\frac{2 \log_3{x} - 4}{\log_3{\left(\frac{x}{9}\right)}} = \log_3{x^7} - \left(\frac{1}{\log_3{9}}\right)^2 - 8
\]

First, let's simplify both sides:

1. The left side simplifies as:
   \[
   \log_3{\left(\frac{x}{9}\right)} = \log_3{x} - \log_3{9} = \log_3{x} - 2
   \]
   Thus:
   \[
   \frac{2 \log_3{x} - 4}{\log_3{x} - 2}
   \]

2. On the right side, since \(\log_3{9} = 2\), we have:
   \[
   \left(\frac{1}{\log_3{9}}\right)^2 = \left(\frac{1}{2}\right)^2 = \frac{1}{4}
   \]
   Therefore:
   \[
   \log_3{x^7} - \frac{1}{4} - 8 = 7\log_3{x} - 8.25
   \]

Thus, the equation becomes:
\[
\frac{2 \log_3{x} - 4}{\log_3{x} - 2} = 7\log_3{x} - 8.25
\]

This equation can be rearranged and solved for \( \log_3{x} \). Let's solve it now.The lowest value of \( x \) that satisfies the given equation is approximately \( x \approx 4.996 \).

Would you like to explore more about this solution or any other related concepts? Let me know if you have any questions or need further clarification.

Here are some related questions you might consider:
1. How does changing the base of the logarithm affect the solution?
2. What would the solution look like if the exponents or coefficients were altered?
3. How can logarithmic equations be applied in real-world scenarios?
4. Could you explain how logarithmic properties were used to simplify the equation?
5. What happens to the solution if the equation is transformed into exponential form?

**Tip:** Remember that logarithmic equations often require careful handling of the domain since logarithms are only defined for positive arguments.

Find the lowest value of x for the equation: \[ \frac{(2 \log_3{x} - 4)}{\log_3{(\frac{x}{9})}} = \log_3{(x^7)} - \left( \frac{1}{\log_3{9}} \right)^2 - 8 \]

Learn how to solve a logarithmic equation involving base 3: \[ \frac{(2 \log_3{x} - 4)}{\log_3{(\frac{x}{9})}} = \log_3{(x^7)} - \left( \frac{1}{\log_3{9}} \right)^2 - 8 \]. This step-by-step solution simplifies complex logarithmic expressions using key logarithmic properties, suitable for advanced high school math students in Grades 11-12.

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