If _105 + _10 ( 5x + 1 ) = _10 ( x + 5 ) \, + 1, then x is equal to :

Answer: Option B

$$\begin{array}{rl}& \Rightarrow {\log }_{10}5+{\log }_{10}\left(5x+1\right)={\log }_{10}\left(x+5\right)+1\\ & \Rightarrow {\log }_{10}\left[5\left(5x+1\right)\right]={\log }_{10}\left[10\left(x+5\right)\right]\\ & \Rightarrow 5\left(5x+1\right)=10\left(x+5\right)\\ & \Rightarrow 5x+1=2x+10\\ & \Rightarrow 3x=9\\ & \Rightarrow x=3\end{array}$$