Question

$\begin{matrix} \log_3 512 & \log_2 3 & \log_8 3\\ \log_4 3 & \log_3 4 & \log_3 4\\ \log_3 8 & \log_4 9 \end{matrix}$ is equal to

• A. 5
• B. 7
• C. 10
• D. 12

Answer 1

There appears to be an error in the question statement or the provided options, as the determinant does not simplify to a constant integer value based on standard calculations. Without clarification or correction of the question, it is not possible to definitively choose one of the integer options.

Answer 2

The determinant is given by

$\Delta = \begin{vmatrix} \log_3 512 & \log_2 3 & \log_8 3\\ \log_4 3 & \log_3 4 & \log_3 4\\ \log_3 8 & \log_4 9 & \log_3 64 \end{vmatrix}$

Using logarithm properties $\log_b a^n = n \log_b a$, $\log_{b^m} a = \frac{1}{m} \log_b a$, and $\log_b a = \frac{1}{\log_a b}$, we can rewrite the entries.

Let $x = \log_3 2$. Then $\log_2 3 = \frac{1}{\log_3 2} = \frac{1}{x}$.

The entries become:

$\log_3 512 = \log_3 2^9 = 9 \log_3 2 = 9x$

$\log_2 3 = 1/x$

$\log_8 3 = \log_{2^3} 3 = \frac{1}{3} \log_2 3 = \frac{1}{3x}$

$\log_4 3 = \log_{2^2} 3 = \frac{1}{2} \log_2 3 = \frac{1}{2x}$

$\log_3 4 = \log_3 2^2 = 2 \log_3 2 = 2x$

$\log_3 8 = \log_3 2^3 = 3 \log_3 2 = 3x$

$\log_4 9 = \log_{2^2} 3^2 = \frac{2}{2} \log_2 3 = \log_2 3 = 1/x$

$\log_3 64 = \log_3 2^6 = 6 \log_3 2 = 6x$

The determinant is:

$\Delta = \begin{vmatrix} 9x & 1/x & 1/(3x)\\ 1/(2x) & 2x & 2x\\ 3x & 1/x & 6x \end{vmatrix}$

Expanding the determinant:

$\Delta = 9x [(2x)(6x) - (2x)(1/x)] - 1/x [(1/(2x))(6x) - (2x)(3x)] + 1/(3x) [(1/(2x))(1/x) - (2x)(3x)]$

$\Delta = 9x [12x^2 - 2] - 1/x [3 - 6x^2] + 1/(3x) [1/(2x^2) - 6x^2]$

$\Delta = (108x^3 - 18x) - (3/x - 6x) + (1/(6x^3) - 2x)$

$\Delta = 108x^3 - 18x - 3/x + 6x + 1/(6x^3) - 2x$

$\Delta = 108x^3 - 14x - 3/x + 1/(6x^3)$

Where $x = \log_3 2$. This expression is generally not an integer and depends on the value of $x$. Given that the options are integers, there is likely an error in the question statement or the provided options, as the determinant does not simplify to a constant integer value based on standard calculations. Without clarification or correction of the question, it is not possible to definitively choose one of the integer options.