Equation with a sum of logarithms

Text

Solve the equation $\log_2(x-1)+\log_2(x+1)=3$.

Solution

Existence conditions: $x-1>0$ and $x+1>0$, i.e. $x>1$. Use the sum rule: $\log_2\big[(x-1)(x+1)\big]=3\;\Rightarrow\; x^{2}-1=2^{3}=8\;\Rightarrow\; x^{2}=9\;\Rightarrow\; x=\pm 3.$ The solution $x=-3$ violates $x>1$ and is discarded. Only $x=3$ remains. $\boxed{\,x=3\,}$