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 (