The equation in the image appears to be:

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

We are tasked with finding the lowest value of $x$.

Let me break down the equation and solve it step by step.

1. Simplify the logarithmic expressions:

   • $\log_3(x^7) = 7 \log_3(x)$.
   • $\log_3 \left( \frac{x}{9} \right) = \log_3(x) - \log_3(9) = \log_3(x) - 2$, since $\log_3(9) = 2$.
   • $\frac{1}{\log_3(3)} = 1$, since $\log_3(3) = 1$, so $\left( \frac{1}{\log_3(3)} \right)^2 = 1^2 = 1$.

2. Substitute into the equation: $\frac{2 \log_3 x - 4}{\log_3 x - 2} = 7 \log_3 x - 1 - 8$ Simplifying further: $\frac{2 \log_3 x - 4}{\log_3 x - 2} = 7 \log_3 x - 9$

3. Multiply both sides by $\log_3 x - 2$ to eliminate the denominator: $2 \log_3 x - 4 = (7 \log_3 x - 9)(\log_3 x - 2)$ Expand the right-hand side: $2 \log_3 x - 4 = 7 (\log_3 x)^2 - 14 \log_3 x - 9 \log_3 x + 18$ Simplifying the right-hand side: $2 \log_3 x - 4 = 7 (\log_3 x)^2 - 23 \log_3 x + 18$

4. Move all terms to one side: $0 = 7 (\log_3 x)^2 - 25 \log_3 x + 22$

This is a quadratic equation in terms of $\log_3 x$. Now, I will solve this quadratic equation using the quadratic formula:

$\log_3 x = \frac{-(-25) \pm \sqrt{(-25)^2 - 4 \cdot 7 \cdot 22}}{2 \cdot 7}$

Let me calculate this next.The solutions for $\log_3 x$ are:

$\log_3 x = 2 \quad \text{or} \quad \log_3 x = 1.5714$

To find $x$, we take the inverse logarithm (base 3) of both values:

• If $\log_3 x = 2$, then $x = 3^2 = 9$.
• If $\log_3 x = 1.5714$, then $x \approx 3^{1.5714} \approx 4.99$.

Thus, the lowest value of $x$ is approximately 4.99.

Would you like more details or further clarification?

Here are 5 related questions for further exploration:

1. How does the change of base formula for logarithms work?
2. What are common properties of logarithmic equations?
3. How can the quadratic formula be derived from a general quadratic equation?
4. Can logarithmic equations have multiple solutions?
5. What are real-world applications of logarithms in various fields?

Tip: When solving logarithmic equations, always check if the solutions fall within the domain of the logarithmic function.