Logarithms

To solve for the determinant of the given matrix: we employ properties of logarithms and determinants. Notice the matrix is constructed using logarithmic relationships derived from distinct bases .
Our aim is to show the determinant equals zero. To simplify, recall these identities:

Substituting gives the matrix:

1
	1
		1

Consider the determinant expansion by the first row:

Each bracket simplifies to zero:

• First:
• Second:
• Third:

Thus, the terms cancel due to the subtraction of equivalent fractions. Therefore, the determinant equals:
Hence, the determinant of the matrix is zero.