3.2: GP sum to infinity and AP of logarithms

A geometric progression has first term $10$ and sum to infinity $6$.

Find the common ratio of this progression.

Hence find the sum of the first $7$ terms, giving your answer correct to $2$ decimal places.

The first three terms of an arithmetic progression are $\log_{x}3$, $\log_{x}3^{2}$, $\log_{x}3^{3}$.

Find the common difference of this progression.

Find, in terms of $n$ and $\log_{x}3$, the sum to $n$ terms of this progression. Simplify your answer.

Given that $S_{n}=3081\log_{x}3$, find the value of $n$.

Given also that this sum to $n$ terms is equal to $1027$, find the value of $x$.