Question
Solve: $\log_5( x + 1 ) - 1 = 1 + \log_5( x - 1 ).$
✓
Answer
$ \log _5(x+1)-1=1+\log _5(x-1)$
$ \Rightarrow \log _5(x+1)-\log _5(x-1)=2$
$ \Rightarrow \log _5 \frac{x+1}{x-1}=2$
$ \Rightarrow \frac{x+1}{x-1}=5^2$
$ \Rightarrow \frac{x+1}{x-1}=25$
$ \Rightarrow x+1=25(x-1)$
$ \Rightarrow x+1=25 x-25$
$ \Rightarrow 25 x-x=25+1$
$ \Rightarrow 24 x=26$
$ \Rightarrow x=\frac{26}{24}$
$=\frac{13}{12}$
$ \Rightarrow \log _5(x+1)-\log _5(x-1)=2$
$ \Rightarrow \log _5 \frac{x+1}{x-1}=2$
$ \Rightarrow \frac{x+1}{x-1}=5^2$
$ \Rightarrow \frac{x+1}{x-1}=25$
$ \Rightarrow x+1=25(x-1)$
$ \Rightarrow x+1=25 x-25$
$ \Rightarrow 25 x-x=25+1$
$ \Rightarrow 24 x=26$
$ \Rightarrow x=\frac{26}{24}$
$=\frac{13}{12}$
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