The equation presented in the image 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 will break down the steps to solve for the lowest value of $x$.

Step 1: Simplify both sides

• First, simplify $\log_3 \left( \frac{x}{9} \right)$. Using the property of logarithms, this becomes:

$\log_3 \left( \frac{x}{9} \right) = \log_3 x - \log_3 9 = \log_3 x - 2$

• The right side also has some simplifications:
  • $\log_3(x^7) = 7 \log_3 x$
  • Since $\log_3 3 = 1$, $\left( \frac{1}{\log_3 3} \right)^2 = 1^2 = 1$

Thus, the equation becomes:

$\frac{2 \log_3 x - 4}{\log_3 x - 2} = 7 \log_3 x - 1 - 8$

Step 2: Combine like terms on the right-hand side

The right-hand side simplifies to:

$7 \log_3 x - 9$

Now the equation is:

$\frac{2 \log_3 x - 4}{\log_3 x - 2} = 7 \log_3 x - 9$

Step 3: Solve the equation

To solve for $\log_3 x$, multiply both sides by $\log_3 x - 2$ to eliminate the fraction:

$2 \log_3 x - 4 = (7 \log_3 x - 9)(\log_3 x - 2)$

Expand the right-hand side:

$2 \log_3 x - 4 = 7 (\log_3 x)^2 - 14 \log_3 x - 9 \log_3 x + 18$

Simplify:

$2 \log_3 x - 4 = 7 (\log_3 x)^2 - 23 \log_3 x + 18$

Step 4: Rearrange the equation into a quadratic form

Move all terms to one side of the equation:

$7 (\log_3 x)^2 - 25 \log_3 x + 22 = 0$

Step 5: Solve the quadratic equation

Now use the quadratic formula:

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

Simplify the discriminant:

$\log_3 x = \frac{25 \pm \sqrt{625 - 616}}{14}$ $\log_3 x = \frac{25 \pm \sqrt{9}}{14}$ $\log_3 x = \frac{25 \pm 3}{14}$

Thus, we have two possible solutions for $\log_3 x$:

$\log_3 x = \frac{25 + 3}{14} = 2 \quad \text{or} \quad \log_3 x = \frac{25 - 3}{14} = \frac{22}{14} = \frac{11}{7}$

Step 6: Solve for $x$

If $\log_3 x = 2$, then $x = 3^2 = 9$.

If $\log_3 x = \frac{11}{7}$, then $x = 3^{\frac{11}{7}}$.

Final Answer:

The lowest value of $x$ is $9$.

Do you have any questions or would you like further clarification?

Here are some related questions to deepen understanding:

1. How does the change of base formula work for logarithms?
2. What are the properties of logarithmic functions that were used in this problem?
3. How do you solve quadratic equations like the one in this problem?
4. What happens when you deal with different bases in logarithms, e.g., base 10 or base 2?
5. How can logarithmic equations be applied in real-life scenarios, such as growth models or sound intensity?

Tip: When working with logarithmic equations, always simplify the logs using properties like product, quotient, and power rules before solving.