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1. If $\log_{10} 5 + \log_{10} (5 x + 1)$ = $\log_{10}$ $(x + 5)$ $+$ $1,$ then x is equal to :

A. 1

B. 3

C. 5

D. 10

Solution:

\begin{aligned} \Rightarrow \log_{10} 5 + \log_{10} (5 x + 1) = \log_{10} (x + 5) + 1 \\ \Rightarrow \log_{10} [5 (5 x + 1)] = \log_{10} [10 (x + 5)] \\ \Rightarrow 5 (5 x + 1) = 10 (x + 5) \\ \Rightarrow 5 x + 1 = 2 x + 10 \\ \Rightarrow 3 x = 9 \\ \Rightarrow x = 3 \end{aligned}

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