((2 log_(3) (x))-4)/( log_(3) ( x/9 )) = log_(3) (x^(7))-( 1/( log_(x) 3) -8

We are tasked with solving the equation:

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

Let's go step by step to simplify and solve this equation.

### Step 1: Simplify the logarithmic terms
Using logarithmic properties, we can simplify the terms:

- \(\log_3\left(\frac{x}{9}\right)\) can be rewritten as:
  \[
  \log_3\left(\frac{x}{9}\right) = \log_3(x) - \log_3(9)
  \]
  Since \(\log_3(9) = 2\), this becomes:
  \[
  \log_3\left(\frac{x}{9}\right) = \log_3(x) - 2
  \]

- \(\log_3(x^7)\) can be rewritten as:
  \[
  \log_3(x^7) = 7 \log_3(x)
  \]

- \(\log_x(3)\) can be rewritten using the change of base formula:
  \[
  \log_x(3) = \frac{1}{\log_3(x)}
  \]

### Step 2: Substitute these into the original equation
We now have the simplified equation:

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

Simplifying the term \(\frac{1}{\frac{1}{\log_3(x)}}\) results in just \(\log_3(x)\). So the equation becomes:

\[
\frac{2 \log_3(x) - 4}{\log_3(x) - 2} = 7 \log_3(x) - (\log_3(x) - 8)
\]

### Step 3: Simplify further
The right-hand side simplifies as:

\[
7 \log_3(x) - (\log_3(x) - 8) = 7 \log_3(x) - \log_3(x) + 8 = 6 \log_3(x) + 8
\]

Thus, the equation becomes:

\[
\frac{2 \log_3(x) - 4}{\log_3(x) - 2} = 6 \log_3(x) + 8
\]

### Step 4: Solve the equation
We now multiply both sides by \(\log_3(x) - 2\) to eliminate the denominator on the left-hand side:

\[
2 \log_3(x) - 4 = (6 \log_3(x) + 8)(\log_3(x) - 2)
\]

Expand the right-hand side:

\[
2 \log_3(x) - 4 = (6 \log_3(x)^2 - 12 \log_3(x) + 8 \log_3(x) - 16)
\]

Simplify the terms:

\[
2 \log_3(x) - 4 = 6 \log_3(x)^2 - 4 \log_3(x) - 16
\]

Move all terms to one side of the equation:

\[
0 = 6 \log_3(x)^2 - 6 \log_3(x) - 12
\]

### Step 5: Solve the quadratic equation
We can divide the entire equation by 6 to simplify:

\[
0 = \log_3(x)^2 - \log_3(x) - 2
\]

This is a quadratic equation in terms of \(\log_3(x)\). Using the quadratic formula:

\[
\log_3(x) = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-2)}}{2(1)}
\]

\[
\log_3(x) = \frac{1 \pm \sqrt{1 + 8}}{2} = \frac{1 \pm \sqrt{9}}{2} = \frac{1 \pm 3}{2}
\]

Thus, we have two solutions for \(\log_3(x)\):

\[
\log_3(x) = \frac{1 + 3}{2} = 2 \quad \text{or} \quad \log_3(x) = \frac{1 - 3}{2} = -1
\]

### Step 6: Find \(x\)
- If \(\log_3(x) = 2\), then \(x = 3^2 = 9\).
- If \(\log_3(x) = -1\), then \(x = 3^{-1} = \frac{1}{3}\).

### Final Solution
The two possible solutions are:

\[
x = 9 \quad \text{or} \quad x = \frac{1}{3}
\]

Would you like further details or have any questions?

### Related Questions:
1. What are the properties of logarithms used in solving equations like this?
2. How does the change of base formula for logarithms work?
3. Can you explain how to solve quadratic equations in more detail?
4. What are some common mistakes to avoid when solving logarithmic equations?
5. How does the quadratic formula apply to logarithmic functions?

### Tip:
Always verify the solutions by substituting them back into the original equation to check for extraneous solutions.

Learn how to solve a complex logarithmic equation that involves simplifying logarithmic expressions and solving a quadratic equation. This step-by-step solution covers key logarithmic properties, the change of base formula, and applying the quadratic formula. Suitable for advanced high school students (Grades 10-12).

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