Solving Logarithmic Equations with Domain Considerations
Consider the equation \(\log_2(x-1) + \log_2(3) = \log_2(2x+1)\). Solve for \(x\) by first applying logarithmic laws, then explicitly solving the resulting equation. Finally, check whether the solution satisfies the domain restrictions implied by the logarithmic functions.
Part a
Apply appropriate logarithmic laws to combine the left side of the equation \(\log_2(x-1) + \log_2(3) = \log_2(2x+1)\) into a single logarithmic expression.