We are given the following equation:

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

Step 1: Simplify the logarithmic expressions

Left side:

$\frac{2 \log_3(x) - 4}{\log_3(x/9)}$

Using the logarithmic property $\log_b \left(\frac{a}{c}\right) = \log_b(a) - \log_b(c)$, we can simplify $\log_3(x/9)$: $\log_3\left(\frac{x}{9}\right) = \log_3(x) - \log_3(9)$ Since $\log_3(9) = 2$, this becomes: $\log_3(x/9) = \log_3(x) - 2$ Thus, the left-hand side becomes: $\frac{2 \log_3(x) - 4}{\log_3(x) - 2}$

Right side:

$\log_3(x^7) - \left(\frac{1}{\log_x(3)}\right)^2 - 8$ Using the logarithmic power rule $\log_b(a^n) = n \log_b(a)$, we get: $\log_3(x^7) = 7 \log_3(x)$

Next, simplify $\frac{1}{\log_x(3)}$. By the change of base formula: $\log_x(3) = \frac{1}{\log_3(x)}$ Thus, $\frac{1}{\log_x(3)} = \log_3(x)$ and $\left(\frac{1}{\log_x(3)}\right)^2 = \log_3(x)^2$

Therefore, the right-hand side becomes: $7 \log_3(x) - \log_3(x)^2 - 8$

Step 2: Set up the equation

The equation is now: $\frac{2 \log_3(x) - 4}{\log_3(x) - 2} = 7 \log_3(x) - \log_3(x)^2 - 8$

Let $y = \log_3(x)$, so the equation becomes: $\frac{2y - 4}{y - 2} = 7y - y^2 - 8$

Step 3: Solve for $y$

Multiply both sides by $y - 2$ to eliminate the denominator: $2y - 4 = (7y - y^2 - 8)(y - 2)$ Expand the right-hand side: $2y - 4 = (7y - y^2 - 8)(y - 2)$ $2y - 4 = 7y^2 - 14y - y^3 + 2y^2 - 8y + 16$ $2y - 4 = -y^3 + 9y^2 - 22y + 16$ Move everything to one side: $0 = -y^3 + 9y^2 - 24y + 20$ $y^3 - 9y^2 + 24y - 20 = 0$

Step 4: Solve the cubic equation

We now solve $y^3 - 9y^2 + 24y - 20 = 0$. Testing small integer values, we find:

For $y = 2$: $2^3 - 9 \cdot 2^2 + 24 \cdot 2 - 20 = 8 - 36 + 48 - 20 = 0$ So, $y = 2$ is a solution.

Step 5: Factor and solve the quadratic

Now factor $(y - 2)$ out of the cubic equation. Using synthetic division, we factor $y^3 - 9y^2 + 24y - 20$ as: $(y - 2)(y^2 - 7y + 10) = 0$ Solving $y^2 - 7y + 10 = 0$ using the quadratic formula: $y = \frac{7 \pm \sqrt{(-7)^2 - 4(1)(10)}}{2(1)} = \frac{7 \pm \sqrt{49 - 40}}{2} = \frac{7 \pm \sqrt{9}}{2}$ $y = \frac{7 \pm 3}{2}$ So, $y = \frac{7 + 3}{2} = 5 \quad \text{or} \quad y = \frac{7 - 3}{2} = 2$

Thus, $y = 5$ or $y = 2$.

Step 6: Find $x$

Since $y = \log_3(x)$, we have: $\log_3(x) = 5 \quad \Rightarrow \quad x = 3^5 = 243$ $\log_3(x) = 2 \quad \Rightarrow \quad x = 3^2 = 9$

Step 7: Conclusion

The possible values of $x$ are 9 and 243. Since we are asked for the lowest value, the lowest value of $x$ is:

$\boxed{9}$

Would you like any further details on the solution?

Related Questions:

1. What is the change of base formula for logarithms?
2. How do you solve cubic equations by synthetic division?
3. How do logarithmic properties simplify equations?
4. What is the significance of the quadratic formula?
5. How can you verify solutions for logarithmic equations?

Tip:

Always verify each solution to ensure it does not lead to a mathematical inconsistency, especially when dealing with logarithms!