Question

If $a = 2024$, $b = 2023$, and $c = 2022$, then the determinant \[ \begin{vmatrix} \log_a a & \log_a b & \log_a c \\ \log_b a & \log_b b & \log_b c \\ \log_c a & \log_c b & \log_c c \end{vmatrix} \] is:

• A. 0
• B. 1
• C. 2024
• D. 2022

Hint:

When dealing with logarithmic determinants, apply the change of base formula to simplify and check for row or column proportionality to quickly find zero determinants.

Answer 1

The Correct Option is A

Step 1: Use the change of base formula

\[\log_x y = \frac{\log y}{\log x},\] where \(\log\) can be any common base (e.g., natural log).

Step 2: Rewrite the determinant entries using this formula:

\[ \begin{vmatrix} \frac{\log a}{\log a} & \frac{\log b}{\log a} & \frac{\log c}{\log a} \\ \frac{\log a}{\log b} & \frac{\log b}{\log b} & \frac{\log c}{\log b} \\ \frac{\log a}{\log c} & \frac{\log b}{\log c} & \frac{\log c}{\log c} \end{vmatrix} = \begin{vmatrix} 1 & \frac{\log b}{\log a} & \frac{\log c}{\log a} \\ \frac{\log a}{\log b} & 1 & \frac{\log c}{\log b} \\ \frac{\log a}{\log c} & \frac{\log b}{\log c} & 1 \end{vmatrix}. \]

Step 3: Let

\[x = \log a, \quad y = \log b, \quad z = \log c.\]
The matrix becomes \[ \begin{vmatrix} 1 & \frac{y}{x} & \frac{z}{x} \\ \frac{x}{y} & 1 & \frac{z}{y} \\ \frac{x}{z} & \frac{y}{z} & 1 \end{vmatrix}. \]

Step 4: Multiply rows by \(x, y, z\) respectively (scaling determinant by \(xyz\))

\[ xyz \times D = \begin{vmatrix} x & y & z \\ x & y & z \\ x & y & z \end{vmatrix}. \]

Step 5: Evaluate the determinant

Since all rows are identical, \[ \det = 0. \]

Step 6: Conclude the value of original determinant

Because \(x,y,z \neq 0\), \[ D = 0. \]