Evaluate the determinant `Delta=|(log_3 512, log_4 3),(log_3 8, log_4 9)|`

Options

A. `15/2`
B. 12
C. `14/3`
D. 6

Answer

To evaluate the determinant

\[ \Delta = \begin{vmatrix} \log_3 512 & \log_4 3 \\ \log_3 8 & \log_4 9 \end{vmatrix} \]

we will follow the steps below:

### Step 1: Calculate the logs
First, we need to express the logarithms in a simpler form.

1. **Calculate \(\log_3 512\)**:
\[ 512 = 2^9 \implies \log_3 512 = \log_3 (2^9) = 9 \log_3 2 \]

2. **Calculate \(\log_4 3\)**:
Using the change of base formula:
\[ \log_4 3 = \frac{\log_3 3}{\log_3 4} = \frac{1}{\log_3 (2^2)} = \frac{1}{2 \log_3 2} \]

3. **Calculate \(\log_3 8\)**:
\[ 8 = 2^3 \implies \log_3 8 = \log_3 (2^3) = 3 \log_3 2 \]

4. **Calculate \(\log_4 9\)**:
\[ 9 = 3^2 \implies \log_4 9 = \frac{\log_3 9}{\log_3 4} = \frac{2}{2 \log_3 2} = \frac{1}{\log_3 2} \]

### Step 2: Substitute the values into the determinant
Now substitute these values into the determinant:

\[ \Delta = \begin{vmatrix} 9 \log_3 2 & \frac{1}{2 \log_3 2} \\ 3 \log_3 2 & \frac{1}{\log_3 2} \end{vmatrix} \]

### Step 3: Calculate the determinant
Using the formula for the determinant of a 2x2 matrix:

\[ \Delta = A \cdot D - B \cdot C \]
where \(A = 9 \log_3 2\), \(B = \frac{1}{2 \log_3 2}\), \(C = 3 \log_3 2\), and \(D = \frac{1}{\log_3 2}\).

Calculating:

\[ \Delta = (9 \log_3 2) \cdot \left(\frac{1}{\log_3 2}\right) - \left(\frac{1}{2 \log_3 2}\right) \cdot (3 \log_3 2) \]

This simplifies to:

\[ \Delta = 9 - \frac{3}{2} = 9 - 1.5 = 7.5 \]

### Step 4: Final simplification
To express \(7.5\) as a fraction:

\[ 7.5 = \frac{15}{2} \]

Thus, the final value of the determinant is:

\[ \Delta = \frac{15}{2} \]