The first term of the G.P. be A and its common ratio be R.
Hence,
\[{{p}^{th}}~term\text{ }=\text{ }a\text{ }\Rightarrow \text{ }A{{R}^{p\text{ }-\text{ }1}}~=\text{ }a\]
\[{{q}^{th}}~term\text{ }=\text{ }b\text{ }\Rightarrow \text{ }A{{R}^{q\text{ }-\text{ }1}}~=\text{ }b\]
\[{{r}^{th}}~term\text{ }=\text{ }c\text{ }\Rightarrow \text{ }A{{R}^{r\text{ }-\text{ }1}}~=\text{ }c\]
Now,
Taking log on both the sides,
\[log(\text{ }{{a}^{q-r}}~x\text{ }{{b}^{r-p}}~x\text{ }{{c}^{p-q}}~)\text{ }=\text{ }log\text{ }1\]
\[\Rightarrow \text{ }\left( q\text{ }-\text{ }r \right)log\text{ }a\text{ }+\text{ }\left( r\text{ }-\text{ }p \right)log\text{ }b\text{ }+\text{ }\left( p\text{ }-\text{ }q \right)log\text{ }c\text{ }=\text{ }0\]
– Hence Proved