If l, m and n are pth, qth and rth terms of a G.P. and are all positive, then loglp1logmq1lognr1 equals

If l, m and n are pth, qth and rth terms of a G.P. and are all positive, then loglp1logmq1lognr1 equals

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If l, m and n are $p^{t h}, q^{t h} a n d r^{t h}$ terms of a G.P. and are all positive, then $|\begin{matrix} \log l & p & 1 \\ \log m & q & 1 \\ \log n & r & 1 \end{matrix}|$ equals

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Detailed Solution

$\begin{array}{l} i = {AR}^{p - 1} \Rightarrow \log i = \log A + (p - 1) \log R \Rightarrow m = A R^{q -} \\ ^{1} \Rightarrow \log m = \log A + (q - 1) \log R \\ \Rightarrow n = A R^{r - 1} \Rightarrow \log n = \log A + (r - 1) \log R \\ n o w, | \begin{matrix} \log l & p & 1 \\ \log m & q & 1 \\ \log n & r & 1 \end{matrix} | = | \begin{matrix} \log A + (p - 1) \log R & p & 1 \\ \log A + (q - 1) \log R & q & 1 \\ \log A + (r - 1) \log R & r & 1 \end{matrix} | \\ = | \begin{matrix} p - 1 & p & 1 \\ q - 1 & q & 1 \\ r - 1 & r & 1 \end{matrix} | (u \sin g C_{1} \to C_{1} - \log A . C_{3}) \end{array}$