Question

If $a,b,c,d$ are positive and are the $p^{th},q^{th},r^{th}$ terms respectively of a G.P. show without expanding that,
$\left|\begin{array}{ccc}loga& p& 1\\ logb& q& 1\\ logc& r& 1\end{array}\right|=0$

Answer 1

Let $A$ be the first term and $R$ be the common ratio of $G.P.,$ then

$a=$ pth terms $=A{R}^{p-1}$

$\Rightarrow \log a=\log A+(p-1)\log R$

$b=$ qth term $=A{R}^{q-1}$

$\Rightarrow \log b=\log A+(q-1)\log R$

$c=$ rth term $=A{R}^{r-1}$

$\log c=\log A+(r-1)\log R$
Consider, $\left|\begin{array}{ccc}\log a& p& 1\\ \log b& q& 1\\ \log c& r& 1\end{array}\right|$
$=\left|\begin{array}{ccc}\log A+(p-1)\log R& p& 1\\ \log A+(q-1)\log R& q& 1\\ \log A+(r-1)\log R& r& 1\end{array}\right|$
$=\left|\begin{array}{ccc}\log A& p& 1\\ \log A& q& 1\\ \log A& r& 1\end{array}\right|+\left|\begin{array}{ccc}(p-1)\log R& p& 1\\ (q-1)\log R& q& 1\\ (r-1)\log R& r& 1\end{array}\right|$
$=logA\left|\begin{array}{ccc}1& p& 1\\ 1& q& 1\\ 1& r& 1\end{array}\right|+\log R\left|\begin{array}{ccc}(p-1)& p& 1\\ (q-1)& q& 1\\ (r-1)& r& 1\end{array}\right|$
$=\log R\left|\begin{array}{ccc}(p-1)& p& 1\\ (q-1)& q& 1\\ (r-1)& r& 1\end{array}\right|$

Applying $C_1\to C_1-logA{C}_3$
$=\left|\begin{array}{ccc}(p-1)\log R& p& 1\\ (q-1)\log R& q& 1\\ (r-1)\log R& r& 1\end{array}\right|$
$=\log R\left|\begin{array}{ccc}(p-1)& p& 1\\ (q-1)& q& 1\\ (r-1)& r& 1\end{array}\right|$

Applying $C_2\to C_2-C_3$
$=\log R\left|\begin{array}{ccc}(p-1)& p-1& 1\\ (q-1)& q-1& 1\\ (r-1)& r-1& 1\end{array}\right|$
$=0$ (Since $C_1$ & $C_2$ identical)