Question

If l, m, n are the pth, qth and rth terms of a G.P. respectively and l, m, n > 0, then

\[ \begin{vmatrix} \log_l p & 1 \\ \log_m q & 1 \\ \log_n r & 1 \end{vmatrix} \]

• A. \( -1 \)
• B. \( 2 \)
• C. \( 1 \)
• D. \( 0 \)

Hint:

When working with logarithms in geometric progressions, recall that the geometric mean relates to the arithmetic mean of the logarithms of the terms.

Answer 1

The Correct Option is C

We are
given that \( l, m, n \) are the \( p \)-th, \( q \)-th, and \( r \)-th terms of a G.P. In a geometric progression, the general form of the \( n \)-th term is:
\[ t_n = a \cdot r^{n-1}, \] where \( a \) is the first term and \( r \) is the common ratio. Now, we are dealing with logarithmic expressions involving these terms. To solve the problem, we use the properties of logarithms and the fact that the terms are in G.P. For a G.P. with terms \( l, m, n \), we have the following relations: \[ m = \sqrt{l \cdot n}, \] since \( m \) is the geometric mean of \( l \) and \( n \). Taking logarithms of both sides: \[ \log m = \frac{1}{2} (\log l + \log n). \] Now, applying the properties of logarithms, we solve the equation: \[ \left| \log l \, p \, 1 \right| = 1. \] Thus, the correct answer is \( 1 \).