Solve the Logarithmic Equation: ln x + ln(x+1) - ln2 = 3

The equation to solve is $\ln x + \ln(x+1) - \ln 2 = 3$.

Step 1: Combine the logarithms using the product and quotient rules:

$\ln (x(x+1)) - \ln 2 = 3 \quad \text{becomes} \quad \ln \left(\frac{x(x+1)}{2}\right) = 3.$

Step 2: Eliminate the logarithm by exponentiating both sides:

$\frac{x(x+1)}{2} = e^3.$

Step 3: Solve for $x$ by clearing the fraction:

$x(x+1) = 2e^3.$

Step 4: Expand and set up a quadratic equation:

$x^2 + x - 2e^3 = 0.$

Step 5: Use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = 1$, $b = 1$, and $c = -2e^3$:

$x = \frac{-1 \pm \sqrt{1^2 - 4 \times 1 \times (-2e^3)}}{2 \times 1}.$

Step 6: Simplify under the square root:

$x = \frac{-1 \pm \sqrt{1 + 8e^3}}{2}.$

Step 7: Ensure $x > 0$. Given $\sqrt{1 + 8e^3}$ will be positive, $\frac{-1 + \sqrt{1 + 8e^3}}{2}$ is the valid solution.

Therefore, the solution to the problem is $\frac{-1+\sqrt{1+8e^3}}{2}$.