Use Newton's method to find an approximate solution of $\ln (x)=5-x$. Start with $x_{0}=8$ and find $x_{2}$

Step 1 ：We are given the equation \(f(x) = \ln(x) + x - 5 = 0\) and we are asked to find the root of this equation using Newton's method. The iterative formula for Newton's method is \(x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\).

Step 2 ：The derivative of \(f(x)\) is \(f'(x) = \frac{1}{x} + 1\).

Step 3 ：We are given that \(x_0 = 8\), and we need to find \(x_2\). This means we need to apply the iterative formula twice.

Step 4 ：First, we find \(x_1\) using the formula: \(x_{1} = x_0 - \frac{f(x_0)}{f'(x_0)}\). Substituting the given values, we get \(x_{1} = 3.484940851840146\).

Step 5 ：Next, we find \(x_2\) using the formula: \(x_{2} = x_1 - \frac{f(x_1)}{f'(x_1)}\). Substituting the values, we get \(x_{2} = 3.6921037570454724\).

Step 6 ：Final Answer: \(x_{2} = \boxed{3.6921037570454724}\)