Step-by-step Solution:

Correct Answer: A (125)

Note on the problem statement: The expression as written, $x^{(8 \log_x(x)) - 24}$, simplifies to $x^{(8(1)) - 24} = x^{-16}$. This would lead to a single value for $x$. Since the question asks for the "product of all possible values of x," it implies there should be more than one solution. This suggests a typo in the base of the logarithm. The most logical correction is that the base of the logarithm should be 5, to match the base on the right-hand side. The solution below assumes the equation is $x^{(8 \log_5(x)) - 24} = 5^{-4}$.

Step 1: Transform the Equation Using Logarithms

We start with the corrected equation:

$$ x^{(8 \log_5(x)) - 24} = 5^{-4} $$

To solve this, we take the logarithm with base 5 on both sides of the equation:

$$ \log_5\left(x^{(8 \log_5(x)) - 24}\right) = \log_5(5^{-4}) $$

Using the logarithm power rule, $\log(a^b) = b \log(a)$, we can bring the exponent down:

$$ (8 \log_5(x) - 24) \log_5(x) = -4 $$

Step 2: Form a Quadratic Equation

The equation now only involves terms of $\log_5(x)$. To make it easier to solve, we can make a substitution. Let $y = \log_5(x)$.

$$ (8y - 24)y = -4 $$

$8y^2 - 24y = -4$
$8y^2 - 24y + 4 = 0$

Divide the entire equation by 4 to simplify:

$$ 2y^2 - 6y + 1 = 0 $$

Step 3: Find the Product of the Values of x

This is a quadratic equation in $y$. Let its two roots be $y_1$ and $y_2$. These roots correspond to the possible values of $\log_5(x)$. So, if $x_1$ and $x_2$ are the possible values of $x$, then:

• $y_1 = \log_5(x_1)$
• $y_2 = \log_5(x_2)$

We are asked to find the product of all possible values of $x$, which is $x_1 x_2$.

Using the properties of logarithms, we know that the sum of the logarithms is the logarithm of the product:

$$ \log_5(x_1 x_2) = \log_5(x_1) + \log_5(x_2) = y_1 + y_2 $$

We can find the sum of the roots ($y_1 + y_2$) of the quadratic equation $2y^2 - 6y + 1 = 0$ using Vieta's formulas (Sum of roots = $-b/a$):

$$ y_1 + y_2 = -\frac{-6}{2} = 3 $$

Therefore, we have:

$$ \log_5(x_1 x_2) = 3 $$

Step 4: Calculate the Final Answer

To find the product $x_1 x_2$, we convert the logarithmic equation back to its exponential form:

$$ x_1 x_2 = 5^3 = 125 $$