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If $x, y, z$ are all positive and are the $p^{\text {th }}, q^{\text {th }}$ and $r^{\text {th }}$ terms of a geometric progression respectively, then the value of the determinant
$\left|\begin{array}{lll}\log x & p & 1 \\ \log y & q & 1 \\ \log z & r & 1\end{array}\right|$ equals
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$\left|\begin{array}{lll}\log x & p & 1 \\ \log y & q & 1 \\ \log z & r & 1\end{array}\right|$ equals
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Let $a$ and $R$ be the first term and common ratio of a GPrespectively.
So, $T_{p}=a R^{p-1}=x$
$T_{q}=a R^{q-1}=y$ and $T_{r}=a R^{r-1}=z$
$\Rightarrow \log x=\log a+(p-1) \log R$
$\log y=\log a+(q-1) \log R$
and $\log z=\log a+(r-1) \log R$
$\therefore\left|\begin{array}{lll}\log x & p & 1 \\ \log y & q & 1 \\ \log z & r & 1\end{array}\right|=\left|\begin{array}{lll}\log a+(p-1) \log R & p & 1 \\ \log a+(q-1) \log R & q & 1 \\ \log a+(r-1) \log R & r & 1\end{array}\right|$
$=\left|\begin{array}{lll}\log a & p & 1 \\ \log a & q & 1 \\ \log a & r & 1\end{array}\right|+\left|\begin{array}{lll}(p-1) \log R & p & 1 \\ (q-1) \log R & q & 1 \\ (r-1) \log R & r & 1\end{array}\right|$
$=\log a\left|\begin{array}{lll}1 & p & 1 \\ 1 & q & 1 \\ 1 & r & 1\end{array}\right|+\log R\left|\begin{array}{ccc}p-1 & p-1 & 1 \\ q-1 & q-1 & 1 \\ r-1 & r-1 & 1\end{array}\right|$
$\left(C_{2} \rightarrow C_{2}-C_{3}\right)$
$=0+0=0 \quad(\because$ two col umns are identical $)$
So, $T_{p}=a R^{p-1}=x$
$T_{q}=a R^{q-1}=y$ and $T_{r}=a R^{r-1}=z$
$\Rightarrow \log x=\log a+(p-1) \log R$
$\log y=\log a+(q-1) \log R$
and $\log z=\log a+(r-1) \log R$
$\therefore\left|\begin{array}{lll}\log x & p & 1 \\ \log y & q & 1 \\ \log z & r & 1\end{array}\right|=\left|\begin{array}{lll}\log a+(p-1) \log R & p & 1 \\ \log a+(q-1) \log R & q & 1 \\ \log a+(r-1) \log R & r & 1\end{array}\right|$
$=\left|\begin{array}{lll}\log a & p & 1 \\ \log a & q & 1 \\ \log a & r & 1\end{array}\right|+\left|\begin{array}{lll}(p-1) \log R & p & 1 \\ (q-1) \log R & q & 1 \\ (r-1) \log R & r & 1\end{array}\right|$
$=\log a\left|\begin{array}{lll}1 & p & 1 \\ 1 & q & 1 \\ 1 & r & 1\end{array}\right|+\log R\left|\begin{array}{ccc}p-1 & p-1 & 1 \\ q-1 & q-1 & 1 \\ r-1 & r-1 & 1\end{array}\right|$
$\left(C_{2} \rightarrow C_{2}-C_{3}\right)$
$=0+0=0 \quad(\because$ two col umns are identical $)$
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