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The problem provides two matrices \( P \) and \( Q \) and asks to find the value of a constant \( c \) under the condition \( \log(|P|) + \log(|Q|) = 5c \), where \( |P| \) and \( |Q| \) denote the determinants of matrices \( P \) and \( Q \) respectively.

### Given matrices:
\[
P = \begin{bmatrix}
4\sqrt{6} & 3 & -2a \\
-4\sqrt{6} & 2a & -3 \\
2a - 3 & 3\sqrt{6} & 4\sqrt{6}
\end{bmatrix}
\]
\[
Q = \begin{bmatrix}
0 & 3\sqrt{3} & \sqrt{2} + 3 \\
8 - 3\sqrt{6} & 0 & 6\sqrt{2} - 3 \\
3\sqrt{2} - 3 & 2b & 0
\end{bmatrix}
\]

The condition provided is:
\[
\log(|P|) + \log(|Q|) = 5c
\]

This simplifies to:
\[
\log(|P| \times |Q|) = 5c
\]

This implies:
\[
|P| \times |Q| = e^{5c}
\]

To solve this, we need to:
1. Compute the determinants \( |P| \) and \( |Q| \).
2. Find the value of \( c \) by substituting these determinants into the equation.

I'll start by calculating the determinants of \( P \) and \( Q \).The determinants of matrices \( P \) and \( Q \) are:

\[
|P| = 8a^3 - 12a^2 + 318a + 531
\]
\[
|Q| = -18\sqrt{6}b - 12\sqrt{3}b + 16\sqrt{2}b + 48b - 81\sqrt{6} + 135\sqrt{3}
\]

Given the condition \( \log(|P|) + \log(|Q|) = 5c \), this simplifies to:

\[
\log(|P| \times |Q|) = 5c
\]

\[
|P| \times |Q| = e^{5c}
\]

Next, I'll determine \( c \) by simplifying the expression for \( |P| \times |Q| \). To do this, I will first calculate \( |P| \times |Q| \) and then solve for \( c \).The product of the determinants \( |P| \times |Q| \) is:

\[
(8a^3 - 12a^2 + 318a + 531) \times \left(-18\sqrt{6}b - 12\sqrt{3}b + 16\sqrt{2}b + 48b - 81\sqrt{6} + 135\sqrt{3}\right)
\]

The value of \( c \) is given by:

\[
c = \frac{1}{5} \log\left(-144\sqrt{6}a^3b - 96\sqrt{3}a^3b + 128\sqrt{2}a^3b + 384a^3b - 648\sqrt{6}a^3 + 1080\sqrt{3}a^3 - 576a^2b - 192\sqrt{2}a^2b + 144\sqrt{3}a^2b + 216\sqrt{6}a^2b - 1620\sqrt{3}a^2 + 972\sqrt{6}a^2 - 5724\sqrt{6}ab - 3816\sqrt{3}ab + 5088\sqrt{2}ab + 15264ab - 25758\sqrt{6}a + 42930\sqrt{3}a - 9558\sqrt{6}b - 6372\sqrt{3}b + 8496\sqrt{2}b + 25488b - 43011\sqrt{6} + 71685\sqrt{3}\right)
\]

This expression for \( c \) is quite complex, and its specific value depends on the values of \( a \) and \( b \).

If you have specific values for \( a \) and \( b \), we can substitute them to find the numerical value of \( c \).

Would you like to proceed with a specific value for \( a \) and \( b \) or need further details on the solution?

---

**Related Questions:**
1. How do we calculate the determinant of a 3x3 matrix?
2. What are the properties of determinants when multiplying matrices?
3. Can the logarithm of a product of determinants be split into a sum?
4. How would the value of \( c \) change if \( a \) or \( b \) were zero?
5. What is the significance of the exponential function in this context?

**Tip:** When working with matrix determinants, always verify the conditions under which the determinant is non-zero to ensure valid logarithmic operations.

Explore how to solve a mathematical problem involving matrix determinants and logarithmic equations. This problem requires calculating the determinants of specific matrices and solving a logarithmic equation involving these determinants. Suitable for advanced undergraduate students and beyond.

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