The sum to n terms of the sequence  $\log \mathrm{a},\log \mathrm{ar},\log {\mathrm{ar}}^2,\dots$ is 

The given sequence can be expressed as

$\log \mathrm{a},(\log \mathrm{a}+\log \mathrm{r}),(\log \mathrm{a}+2\log \mathrm{r})\dots$

which is clearly an A.P. whose first term is log a and 
common difference is log r.

The nth term $=\log \mathrm{a}+(\mathrm{n}-1)\log \mathrm{r}$

Since sum to n terms $\mathrm{s}=\frac{\mathrm{n}}{2}\left({\mathrm{a}}_1+{\mathrm{a}}_{\mathrm{n}}\right)$

$\therefore {\mathrm{S}}_{\mathrm{n}}=\frac{\mathrm{n}}{2}[\log \mathrm{a}+\log \mathrm{a}+(\mathrm{n}-1)\log \mathrm{r}]$

$=\frac{\mathrm{n}}{2}[2\log \mathrm{a}+(\mathrm{n}-1)\log \mathrm{r}]=\frac{\mathrm{n}}{2}\log {\mathrm{a}}^2{\mathrm{r}}^{\mathrm{n}-1}$