Question
If $\log (a + b) = \log a + \log b,$ find $a$ in terms of $b.$
✓
Answer
$ \log (a+b)=\log a+\log b$
$ \Rightarrow \log (a+b)=\log a b$
$ \Rightarrow a+b=a b$
$ \Rightarrow a-a b=-b$
$ \Rightarrow-a b+a=-b$
$ \Rightarrow-a(b-1)=-b$
$ \Rightarrow a(b-1)=b$
$ \Rightarrow a=\frac{b}{b-1}$
$ \Rightarrow \log (a+b)=\log a b$
$ \Rightarrow a+b=a b$
$ \Rightarrow a-a b=-b$
$ \Rightarrow-a b+a=-b$
$ \Rightarrow-a(b-1)=-b$
$ \Rightarrow a(b-1)=b$
$ \Rightarrow a=\frac{b}{b-1}$
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