Question
If $${\log _{10}}5 + {\log _{10}}\left( {5x + 1} \right)$$ = $${\log _{10}}$$ $$\left( {x + 5} \right)$$ $$\, + $$ $$1,$$ then x is equal to :
Answer: Option B
$$\eqalign{
& \Rightarrow {\log _{10}}5 + {\log _{10}}\left( {5x + 1} \right) = {\log _{10}}\left( {x + 5} \right) + 1 \cr
& \Rightarrow {\log _{10}}\left[ {5\left( {5x + 1} \right)} \right] = {\log _{10}}\left[ {10\left( {x + 5} \right)} \right] \cr
& \Rightarrow 5\left( {5x + 1} \right) = 10\left( {x + 5} \right) \cr
& \Rightarrow 5x + 1 = 2x + 10 \cr
& \Rightarrow 3x = 9 \cr
& \Rightarrow x = 3 \cr} $$
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$$\eqalign{
& \Rightarrow {\log _{10}}5 + {\log _{10}}\left( {5x + 1} \right) = {\log _{10}}\left( {x + 5} \right) + 1 \cr
& \Rightarrow {\log _{10}}\left[ {5\left( {5x + 1} \right)} \right] = {\log _{10}}\left[ {10\left( {x + 5} \right)} \right] \cr
& \Rightarrow 5\left( {5x + 1} \right) = 10\left( {x + 5} \right) \cr
& \Rightarrow 5x + 1 = 2x + 10 \cr
& \Rightarrow 3x = 9 \cr
& \Rightarrow x = 3 \cr} $$
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