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To solve the equation given in the image, \( (2 \log_3 x) - 4 = \frac{\log_3 (x^7)}{\log_3 (\frac{x}{9})} - \left(\frac{1}{\log_3 x}\right)^2 - 8 \), we'll simplify and manipulate the expressions:

1. **Change of Base and Simplifications:**
   \[
   \frac{\log_3 (x^7)}{\log_3 (\frac{x}{9})} = \frac{7 \log_3 x}{\log_3 x - \log_3 9} = \frac{7 \log_3 x}{\log_3 x - 2}
   \]
   The term \(\frac{1}{\log_3 x}\) is rewritten as \((\log_3 x)^{-1}\).

2. **Substitution and Rearrangement:**
   Let \( y = \log_3 x \). Then the equation transforms into:
   \[
   2y - 4 = \frac{7y}{y - 2} - y^{-2} - 8
   \]

3. **Common Denominator and Solving the Equation:**
   \[
   2y - 4 = \frac{7y}{y - 2} - \frac{1}{y^2} - 8
   \]
   Multiplying through by \((y - 2)y^2\) to clear the fractions:
   \[
   (2y - 4)y^2(y - 2) = 7yy^2 - (y - 2) - 8y^2(y - 2)
   \]
   \[
   2y^3 - 4y^2 - 4y + 8 = 7y^2 - (y^2 - 2y) - 8y^3 + 16y^2
   \]
   \[
   2y^3 - 4y^2 - 4y + 8 = -y^3 + 22y^2 - 2y - 2
   \]
   Simplifying further to create a polynomial equation:
   \[
   3y^3 - 26y^2 + 2y + 10 = 0
   \]

4. **Finding the Roots:**
   This cubic equation can be solved either through numerical methods or by estimating roots. We can use synthetic division or approximation techniques to find the roots of this equation.

Let's proceed by solving the cubic equation for \( y \) (which is \( \log_3 x \)) and then finding the smallest value of \( x \) from these solutions.The solutions for \( y = \log_3 x \) include complex numbers, indicating that we may have to look for the real roots more carefully. Since we're seeking the smallest value of \( x \), we need to identify the real solution(s) among these roots and then convert this value back to \( x \) using \( x = 3^y \).

I will now filter out the real root(s) and calculate the corresponding value of \( x \) for each real root.The real roots for \( y = \log_3 x \) are approximately 0.69, -0.57, and 8.54. These correspond to the values of \( x \) as approximately 2.13, 0.54, and 11913.13 respectively.

The smallest value of \( x \) from these is approximately 0.54, which is obtained when \( y \approx -0.57 \).

Would you like more details or have any questions about this solution?

Here are some additional topics you might consider:

1. How the change of base formula is applied to logarithmic expressions.
2. The general approach to solving cubic equations analytically.
3. How imaginary roots occur in equations and what they represent.
4. The significance of finding the lowest value in practical scenarios.
5. An alternative approach using numerical methods to find roots.

Tip: In solving complex equations involving logarithms, always consider simplifying the equation first and verifying if the solutions make sense in the context of the problem.

Find the lowest value of X from the given logarithmic equation.

Solve the logarithmic equation step by step to find the lowest value of X using properties of logarithms, change of base formula, and cubic equations. Suitable for advanced high school and college students.

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