Solve for \( x \) : \( \log (x+3)-\log (x+1)=2 \) \( x=\square \) You may enter the exact value or round to 4 decimal places. Question Help: Video Submit Ouestion

To solve the equation \( \log (x+3) - \log (x+1) = 2 \), we can apply the properties of logarithms. Specifically, we can use the quotient rule which states that \( \log a - \log b = \log \left(\frac{a}{b}\right) \). This simplifies our equation to: \[ \log \left(\frac{x+3}{x+1}\right) = 2 \] Next, we can rewrite this logarithmic equation in its exponential form, resulting in: \[ \frac{x+3}{x+1} = 10^2 = 100 \] Now we can cross-multiply: \[ x + 3 = 100(x + 1) \] Expanding the right side gives: \[ x + 3 = 100x + 100 \] To isolate \( x \), we move \( x \) to one side and constants to the other: \[ 3 - 100 = 100x - x \] \[ -97 = 99x \] Now, divide both sides by 99: \[ x = -\frac{97}{99} \approx -0.9798 \] Thus, the solution is: \( x \approx -0.9798 \)