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The given equation is:

\[
\frac{(2 \log_3 x) - 4}{\log_3 \left( \frac{x}{9} \right)} = \log_3(x^7) - \left( \frac{1}{\log_3 3} \right)^2 - 8
\]

We are tasked with finding the lowest value of \( x \).

Let's break down and solve this step by step:

### Step 1: Simplify the denominator on the left side
The denominator of the left-hand side is:
\[
\log_3 \left( \frac{x}{9} \right)
\]
We can apply the logarithmic property:
\[
\log_3 \left( \frac{x}{9} \right) = \log_3 x - \log_3 9
\]
Since \( \log_3 9 = \log_3 (3^2) = 2 \), this simplifies to:
\[
\log_3 x - 2
\]
So the left-hand side becomes:
\[
\frac{2 \log_3 x - 4}{\log_3 x - 2}
\]

### Step 2: Simplify the right-hand side
On the right-hand side, the term \( \left( \frac{1}{\log_3 3} \right)^2 \) simplifies, because \( \log_3 3 = 1 \). Thus, \( \frac{1}{\log_3 3} = 1 \), and squaring it gives:
\[
1^2 = 1
\]
So, the equation on the right-hand side becomes:
\[
\log_3(x^7) - 1 - 8
\]
Using the logarithmic property \(

(2 log₃ x) - 4 / log₃ (x / 9) = log₃ (x⁷) - (1 / log₃ 3)² - 8. Find the lowest value of X.

Explore how to solve the logarithmic equation (2 log₃ x - 4) / log₃ (x / 9) = log₃ (x⁷) - (1 / log₃ 3)² - 8. The solution demonstrates key logarithmic properties and exponents, suitable for students in Grades 10-12.

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