$\text{ If the sum of the first }20\text{ terms of the series }{\log }_{\left(7^2\right)}x+{\log }_{\left(7^3\right)}x+{\log }_{\left(7^4\right)}x+\dots ..\text{ is }460\text{ , then }x\text{ is }$equal to

The given equation is 

$\begin{array}{l}2{\log }_{7}x+3{\log }_{7}x+4{\log }_{7}x+\dots ..=460\\ \left(20\text{ terms }\right)\\ \Rightarrow (2+3+4+\dots \dots ){\log }_{7}x=460\\ \Rightarrow \left(\frac{21\times 22}{2}-1\right){\log }_{7}x=460\end{array}$

${\log }_{7}x=2\Rightarrow x=7^2$