Solve for $x \cdot \ln (x-2)-3 \ln 2=2$

Step 1 ：We are given the equation \(x \cdot \ln (x-2)-3 \ln 2=2\). This is a transcendental equation, which means it cannot be solved analytically using elementary functions. However, it can be solved numerically.

Step 2 ：We will use the Newton-Raphson method to solve this equation. The Newton-Raphson method is a root-finding algorithm that uses the first few terms of the Taylor series of a function to find its roots.

Step 3 ：The general formula for the Newton-Raphson method is \(x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\), where \(f(x)\) is the function we want to find the root of, and \(f'(x)\) is its derivative.

Step 4 ：In this case, our function is \(f(x) = x \cdot \ln (x-2)-3 \ln 2 - 2\).

Step 5 ：First, we need to find its derivative. The derivative of \(f(x)\) is \(f'(x) = \ln(x-2) + 1\).

Step 6 ：Then, we can use the Newton-Raphson method to find the root of the function. We start with an initial guess for the root, and then iteratively apply the Newton-Raphson formula until we reach a solution that is accurate enough.

Step 7 ：Let's start with an initial guess of \(x_0 = 3\).

Step 8 ：By applying the Newton-Raphson method, we find that the root of the equation \(x \cdot \ln (x-2)-3 \ln 2=2\) is approximately 4.48.

Step 9 ：Final Answer: The solution to the equation is \(\boxed{4.48}\).